It can be either countable or uncountable. You could be having a tea, meaning you're having a cup of tea.
No, it's not that it always has a tail when describing tea. That would be a really really weird grammar rule.
"green tea" is a noun phrase. A noun phrase should always be considered as a whole. As the verb 'pić' (to drink) takes Accusative, this whole noun phrase takes Accusative: zieloną herbatę.
If "green tea" was the subject of the sentence (Green tea is very tasty), it would take Nominative. "Zielona herbata jest bardzo smaczna".
Easiest to check on Wiktionary :) https://en.wiktionary.org/wiki/zielony#Declension_2
nijaki is "zielone", as for neuter Accusative is identical to Nominative, and for męski it matters whether it's żywotny (animate) or nieżywotny (inanimate). The first one is "zielonego" and the latter "zielony" as it's also identical to Nominative.