There is such a thing as a voluntary Tefilla, but we try not to do it since it can be Levatala if it isn’t done right. There is no need to pretend that you are Davenning. You may sit unless someone near you is Davenning Shmone Esrei, in which case you can learn standing.
The only time you are required to pretend is if the Tzibur is saying Shema when you are holding after Barchu, in middle of a Bracha. In that case you put your hand over you eyes and say the words that you are up to, in the tune of Krias Shema, so as not to stand out as not being Mekabel Hashem’s kingdom along with Klal Yisroel.
Aleinu is not like Kaddish and Kedusha, where the idea is to answer. Aleinu, like Krias Shema, is recited along with the Tzibur so as not to seem to disagree.
As far as I know, you don’t say along Aleinu if you are in middle of anywhere else in Davenning, and I think even learning.