mik5
I never heard of saying a voluntary prayer in a situation where it wasnt a question of being necessary, like forgetting Yaaleh vyavo by mincha and saying 2 maarivs that night when it is no longer Rosh chodesh, or other such cases. It seems like it is only an option in such cases. In your case where you certainly Davened already I dont believe theres such a thing as Davening an extra shmoneh esrei just as a voluntary Tefillah.
I would say ask a Rabbi.
“I just don’t feel comfortable standing and pretending that I’m davening when I’m not.”
Who said anything about being reuired to stand? You dont have to stand you may sit.
“Can I answer half-kaddish even if I didn’t say ashrei?”
You may answer to Kaddish and answering to kaddish has nothing to do with ashrei. If you walk by a Shul and from the open window you hear kaddish being recited, although you arent obligated to answer, you may respond amein and its a good thing to do so.
“How can I answer kaddish if I didn’t say tachanun?”
See my previous answer.
“And why am I required to say aleinu?”
Not sure why but I think it is because the Tefillah starts off “Aleinu Leshabeich” It is upon us (our obligation) to praise, so you dont want to show like you are taking yourself out of the group. I will see if I can find something more on the topic.