Not a pun. A pun is word play in the usage of similar sounds / the use of homonyms. For instance, a Freudian pun would take one word and make it sound like a mother.

IIUC, also not a pun. A pun requires the statement to work on multiple levels -- not merely be associated with multiple homonymic meanings -- such that a listener who missed the additional interpretation wouldn't find it nonsensical. (Unless, I suppose, you mean your mother just "sounds like" the words of a Freudian pun?)

My favorite that works both ways: why can't a bicycle stand up without a kickstand? because it's "too tired."

When I say it 'works' both ways, I don't mean that you 'get it' when you say "oh, because a bike has two tires." I mean, if you only heard me as saying 'two-tired,' then that is also literally the reason why a bike cannot stand up on its own. Still makes sense by the setup. See also: "no pun in ten did"

I'm no authority and certainly don't have your credentials, but I've been berated in every-which-way on reddit for this and had to do some of my own research. A quick googling doesn't indicate that there's such a thing as a "Freudian pun," and from the definition I've heard, your "non-pun double entendre" was the closest thing to a pun in your comment.