The taraweeh prayer during Ramadan is a confirmed sunnah of the prophet, but it is not considered obligatory. If you missed it then you missed it, but there is no sin on you for that and there is no need to make it up later.
It sounds to me like you already know how to calculate zakat.
Zakat is due on whatever balances you have maintained for a year or longer at the time of payment, as long as you have reached the nisab. Your first concern, that whatever salary you received a month later wouldn't be zakated for another two years is true, but that's fine.
For example, let's say ...
No, your fast did not break. Your action of drinking happened before Athaan. The moisture just happened to be there afterwards. Also, you could eat after the athaan as the Siyaam starts a bit before dawn, and the athaan is still before that time. Anyway, any unintentional eating drinking is forgiven, even if you had it in the middle of the Siyaam.
Yes, you may. Taraweeh is only a Sunnah Mu‘akkadah, and Witr's status is high enough that the Hanafi madhab took it to be as wajib (obligatory). If you were to only offer one, then Witr would take precedence.
Inshaallah your fast will still be valid. The hadath (impurities) including farts, urinating, janabah, etc. are not the requirement for doing starting the fast. Those are the requirement for other obligation such as the five prayer.