The following is to the best of my knowledge, so anybody feel free to correct me if I’m wrong
If I were mean, I’d say your first question is wrong There’s no such thing as “a kanji for a reading”. There are roughly 100 sounds in the Japanese language (unvoiced, voiced etc.). On the other hand, there are more than 2.000 commonly used kanji, and more than 40.000 kanji overall. So you just can’t have a separate sound for each kanji. Many times you’ll find kanji that are read identically. For example, 「工」 and 「口」 can both be read 「こう」.
To make things even more complicated, many kanji have more than one reading. 「工」 can be read as 「こう」 or 「く」, depending on which word the kanji’s in. 「口」 can be read as 「こう」 or 「く」 or 「くち」.
To come back to your question, 「乾」 and 「渇」 are different kanji with related meanings that happen to have the same reading, too. The verb 「渇く」 means, according to Jisho.org, “to be thirsty” or “to thirst, to crave for something”. The verb 「乾く」 means, according to Jisho.org, “to get dry”. Both are read as 「かわく」.
So 「喉が渇いています」 and 「喉が乾いています」 both mean something like “(my) throat is dry” or “(my) throat is thirsty”, or, as we would say in English, “(I’m) thirsty” or “being thirsty”. And they happen to be read exactly the same.
But since 「乾いています」 is taught in the course as “to be dry” and according to Jisho.org 「喉が乾く」 is a (standing?) phrase for “being thirsty”, it might be a good idea to use 「喉が乾いています」 in the course. That would also match the principle we’ve seen with 「お腹」 and 「お腹が空いてます」. What do you think, @MarikoMiz?
And concerning the second question, 「好きです」 can used to express that one likes anything or anybody, not just food. Judging by the manga and anime I’ve read or seen, this phrase is actually used to confess your feelings to somebody (though I’m not sure if in this case the relatively formal 「~です」 form would be used).