I always look at tüsszent and think of the French verb tousser "to cough" :P
I always think about tissues.
Te can be omitted right?
Sure, yes. In that case a becomes az, since the next word, unokabátyád, begins with a vowel.
I can't access the tips in the middle of a lesson: Is "unokatestvér" a girl cousin and "unokabátya" a boy cousin?
my advice for these:
use unokatestvér for all cousins
never use unokabáty.