Learn Icelandic: Ég tala en þú hlustar ekki. - I speak but you do not listen.

Both are Group 1 verbs (their dictionary forms are að tala and að hlusta). In the present tense, Group 1 verbs take the ending -a for I (ég) and -ar for you (þú).

If I wanted to say "he listens" or "she speaks", what endings would I use?

You would use the exact same -ar ending that you use for you (þú). For he, she, and it (hann, hún, það), Group 1 verbs always end in -ar. So you would say hann hlustar (he listens) and hún talar (she speaks).

Why is ekki at the very end of the sentence instead of before the verb like in English?

In Icelandic, the word for not (ekki) generally goes immediately after the verb it is negating. In the phrase þú hlustar ekki, the verb is hlustar, so ekki must follow it directly.

Since the verbs have different endings, can I drop ég and þú from the sentence?

No, you generally cannot drop subject pronouns in Icelandic. Even though the verb endings change depending on the person, you still must include words like ég (I) and þú (you) for the sentence to be grammatically complete.