If one finishes shmone esrei when the tzibur is already up to aleinu, is he supposed to say aleinu with them or say tachanun first since it is connected to shmone esrei?
This is a tough question. On one hand the main power of tachanun is when it is adjacent to Shemona Esrei, but on the other hand while everyone is standing and saying Aleinu and bowing, it is not derech eretz to not say along with them. Therefore, there are different opinions what you should do in this situation, to say it or not to. Some poskim suggest that you say just the beginning of Aleinu until after the bowing part with the tzibbur, and then say tachanun. This way you acknowledged what the tzibbur is saying and you didn’t make that long of a hefsek.
The M:B 131-1 brings the Magen Avrohom who says that it is permitted to talk out for a davar shbikidusha, as long as it isn’t talking mndane things that will divert his attention. Therefore it would be permitted to say aleinu. However other poskim (Kaf Hachyim 131-2, Sharei Teshuva) say that we may only talk out to answer a davar shbikidusha. See Knei Besem 2-8, who says not to say it. Also see Minchas Yitzchok 9-8(2) not to answer aleinu in middle of pesukei dzimra. R’ Y. Cahn shlit”a.