“Drogerie” is not a synonym for “Apotheke”.
“Apotheke” is a chemist’s (UK)/pharmacy (US), where you can buy medicine that may only be sold in such establishments as well as medicine that is only available on a doctor’s prescription.
“Drogerie” is a place where you buy things such as shaving cream, plasters (band-aids), nappies (diapers), or sunscreen. I think US “drugstore” may fit.
Chemie can refer not only to the school subject but also to “chemistry” between two people (die Chemie zwischen den beiden stimmt nicht).
I think ärgern should be split up into sich ärgern (to get annoyed) and ärgern (to bother, anger). I’m not sure whether “fret” works well for either sense – it’s not transitive for me (“He keeps fretting me”?) but the intransitive sense for me is more like “worry” than “get angry or annoyed”.
abonnieren “to subscribe” might be better as “to subscribe to”, to remind the student that you eine Zeitschrift abonnieren and not abonnieren *zu einer Zeitschrift.
der Cousin is specifically a male cousin, while “cousin” in English can be either. Do you have die Cousine as an accepted alternative? If not, perhaps the clue should be “the (male) cousin” – and you could consider adding die Cousine ; die Kusine as a separate entry.
Is it worth adding der Euro = the euro?
die Ferien is in my experience specifically for the free time that children at school or university have, so “school holidays” might be an appropriate hint which would also help distinguish it from Urlaub, which can be the free time that an employee has as well as for a journey undertaken during free time (in den Urlaub fahren, to go on holidays).
I take it you speak UK English? Perhaps the hint “gas” could be changed to “gas (gaseous substance, not liquid car fuel)” to help potential US “clients” of your course.
For Gegenstand, I think “object” or “item” might be better; I’d say those meanings are rather more common than “cause, subject matter (of an investigation etc.)”.
Gehalt has “salary, wage”; I thought “wage” is usually something paid hourly (= Lohn) while “salary” is monthly (= Gehalt). At least, that’s how I understand the English and German words. Einkommen is “income” in general, whether you get paid hourly or monthly.
gerade is also “straight”.
gratis has the hint “synonym for kostenlos” but there is also umsonst with that meaning, so this may not be sufficiently unambiguous. (umsonst can also mean “in vain”, of course.)
herunter implies not only movement, but specifically movement towards the speaker – movement down and away from the speaker is hinunter.
“the Internet, the interview” should perhaps include the definite article “the”, for consistency with other nouns. (And I would personally capitalise “the Internet” since it’s not just any old internet linking separate networks together, it’s the best-known one that links the entire world.)
The L words also seem to be missing “the” in the English hint.
jemals is not just “one day (in the future)”, it’s also used for “ever (in the past)”: Hast du je(mals) Hundefleisch gegessen? Bist du jemals in New York gewesen? as well as “ever (in the future)”: Ich weiß nicht, ob ich ihn jemals wiedersehe.
Perhaps “ever” is a better translation than “one day”.
der Kinderwagen is, in my experience, mostly for baby carriages / prams where the baby lies on its back. If it sits up (buggy, stroller), I’d say it’s more likely to be der Buggy. YMMV.
das Konsulat is not “the consultant” (person who gives advice) but “the consulate” (building for diplomatic matters).
sich kümmern should perhaps be sich kümmern um to show the typical preposition. And I’m not sure whether “care (for something)” is best – it’s often used to “care for someone (a person), take care of someone” as well, though not in the sense “care for someone = have feelings for someone”.
künstlerisch is missing the -s- in the ending -isch.
der Kunststoff is one word for plastic; das Plastik is another, so just “plastic” is not really a sufficient hint.
In my experience, Plastik is the more commonly-used word in daily speech, while Kunststoff (as the German word) is the more “bureaucratic” word. So it’s worth recognising but one should probably know both and perhaps use Plastik oneself.
…ah, I see you include that as well.
So der Kunststoff should perhaps get “(syn. for “das Plastik”)” added.
links might have hint “to the left, at the left” to distinguish from “left, remaining = übrig”.
öffnen has the hint “(o word)” attached to the German, not the English.
die Party = “the party” – might need disambiguation, since “the party” can also be die Partei for a political party or for “party of the first part” etc. in contracts.
Perhaps “the party, celebration”?
perfect might need a hint “(p word)” to distinguish from vollkommen which is also “perfect”.
Das Rathaus has an unexpected capital on the article
rein is not only “pure” but is also a frequent abbreviation of hinein and herein = in(to). For example, ich gehe rein or da passt nichts mehr rein.
(Though if the questions are usually asked as “given this English phrase, what is the German?” perhaps this additional meaning need not be included? It’s still a very useful word to know, in my opinion.)
die Sahne = the cream – does this need an ambiguator that this is about cream made out of milk? It doesn’t apply to the cream that one might spread on one’s skin, for example (= die Creme, die Crème, die Krem).
schenken - may I suggest “to give as a present”?
schmal - I don’t think “small” is a good translation for this; it’s a false friend even if they have the same origin (like Gift and “gift”).
aufschreiben - I would translate this as most simply as “write down”.
I’m not sure whether “inscribe” works (can you give me an example sentence?), and “record” could be confused for an audio or sound recording but this sense of the verb is aufnehmen. You might “record someone’s time in a race on a form” with aufschreiben but then “write down” will also work, I think.
die Schülerin has a hint in English but der Schüler does not.
When watering, would the learner be expected to remember that the lack of a hint is the hint for the male form?
Also, this is specifically for someone who attends primary or secondary education, a pupil – not a student at a university (= der Student, die Studentin or nowadays sometimes der Studierende, die Studierende).
die Sekretärin is explicitly a female secretary; “the secretary” could also be a male one.
I don’t think sparen and speichern are synonyms at all.
They both translate to “save” in English but with rather different meanings.
sparen is to save money, to spend less than you might otherwise have, or to save money by putting some away for a rainy day.
speichern is to save computer data by writing it to a file.
You can’t Geld speichern (well, maybe if you save it to an electronic wallet?) or eine Datei sparen.
starten is present twice.
Stau could perhaps have a hint “traffic jam” in addition to or instead of just “jam”.
die Straßenbahn is there twice: once under B as in Bahn and once under S as in Straßenbahn.
Ditto with U-Bahn under B and under U.
das Team and die Mannschaft are both glossed simply as “the team” - these should perhaps be disambiguated, e.g. “not “Team”” or “M word” or whatever.
der Tierpark and der Zoo are both glossed simply as “the zoo”.
I’m not sure whether “Tierpark” has to be taught – in my opinion, it’s another “recognising is enough” word.
der Tipp = “the tip, hing” – what’s a “hing”? Typo for “hint”?
das Top = “the top” – in my experience, only for the article of clothing, so perhaps this should be included in the hint. The top of an object is die Oberseite (and “on top” = oben), and the child’s spinning toy is der Kreisel.
die Überstunde is one hour of overtime. Overtime in general is die Überstunden, plural. To work overtime is Überstunden machen.
Several Ü words have hints with “Ü word”, but Ö and Ä words (ärgern, öffnen) have hints “A word, O word”. This should perhaps be unified to “Ä,Ö,Ü word” with umlaut rather than “A,O,Ü” with umlaut sometimes and sometimes not.
die Umgebung and die Umwelt can be synonymous, but I’d say the most common meanings are Umgebung = surroundings and Umwelt = environment (as in natural environment, “save the environment”, “environmentally friendly” etc.). For a learner who wants “absolutely essential vocabulary”, I think one can initially ignore the “surrounding world” meaning of Umwelt, and “environment” as a hint for Umgebung may be misleading. Maybe “environs” would work if you want to use that root without making one think specifically of nature.
Also, the hint for Umwelt spells Umgebung wrong as Umbebung (which sounds like “surrounding by earthquakes” 
)
die Unterstützung is glossed merely as “the support”, which collides a bit with die Förderung which also has “the support”, though with a hint.
verhalten “to restrain, contain” – I’m not sure whether it can mean that. Nor am I sure whether this verb exists as a non-reflexive verb.
Perhaps sich verhalten “to behave (in a certain way)” is what was intended here?
vertreten could perhaps also be glossed “substitute (for)”
der Verwandte is specifically a male relative.
vorstellen: I’m not sure whether “suggest” is a good gloss. Do you have an example sentence?
On the other hand, “present” might be a good gloss – perhaps instead of “represent”, which I’m also unsure about.
wann = “when” – this gloss should be disambiguated to make sure it’s distinguished from wenn and als.
It’s specifically the question word, meaning “at what time?” and not, for example, “when I saw him / when I see him”.
widersprechen, wieder, wiederholen, wiegen seem to be missing English glosses.
(I suggest: widersprechen “to contradict; to answer back” - wieder “again” - wiederholen “to repeat” - wiegen “to weigh, to have a weight of”.)
You have wie “like, as”. Do you also want to include wie “how”?
You have wie viel “how much”. Do you also want to include wie viele “how many”?
wirken = “appeal” - this gloss seems like a mistake to me.
Perhaps “have an effect” could be added.
die Realität and die Wirklichkeit are both glossed simply “the reality”.
wohl = “probably” - this could be confused with wahrscheinlich.
I’m not sure whether wohl has a good translation into English – “must” might work but the constructions are rather different (Du bist wohl verrückt = You must be crazy; Dich hat wohl der wilde Affe gebissen = A wild money must have bitten you).
There’s also the meaning “well; comfortable” as in lebe wohl or ich fühle mich wohl.
worüber is glossed “what … about” and worum, “about what” – both have a hint “one word “W”” but this is probably not helpful in this context.
Perhaps these could be “about what, over what” and “about what, around what” respectively?
Since “over what, around what” are the literal (spatial) meanings from which the metaphorical ones derive.
wundern should perhaps be sich wundern if you gloss it “marvel, wonder”.
The non-reflexive verb is more like “make someone wonder” (Das wundert mich).
das Zeugnis: I wonder whether adding “report card” might be a good idea? This is probably the most common usage of that word, I would think.
“testimony” (in court) I would say is a Zeugenaussage and “evidence” is Beweis, Beweise, Beweismittel.
On the other hand, if you’re LDS, then Zeugnis = “testimony” is a useful word to know…
die Zinsen = “the interest” - should perhaps note that this is interest as in “money, percentage”, not as in “of interest to me”.
You have both zirka and circa.
Those are simply two spellings of the same word; the word should probably just be there once.