am is time. the t- is there because it's a masculine noun beginning with a vowel following the definite article.
Orthography and convention. Basically, you use a hypen between it if lowercase, and none if uppercase.
An t-am, but an tAontas Eoprach
do these phrases need a period at the end of them if they aren't full sentences ?
Just wondering, how would you ask someone 'do you have the time?' Would it be: An bhfuil t-am agat?
Without a definite article, An bhfuil am agat? means "Do you have time?" (to do this for me)
An bhfuil an t-am agat? - "Do you have the time?". Just as in English, it can mean "Do you know what time it is?" or "Do you have enough time to do that task?"