I've always thought of it as getting near to, as in making that push towards a goal or destination. That team is making their push towards the playoffs or making that final push to finish my homework. Probably wrong, but that's my 2 3/4 cents.

Nearly that age

a) I have similar issues with the Scottish "back of [time]" (eg, "I'll be there the back of 4"). I *think* it means a bit past the hour, but I'm not 100% sure!

Never heard this one before, but I’d also immediately understand it as you do. If I say something is at the back of a door, it means it’s past the door, on the other side. One could say they were “back from something (an experience) changed”, which would mean they changed after the thing.

Definitely a.

My biggest language question that I don’t know I’ll ever get a satisfactory answer is: why do so many English speakers, from so many dialects and accents, simply INVENT an “R” sound between some specific words that don’t have any Rs? Most popular in “the ideaR of”. But lots of sentences!

Intrusive R! en.m.wikipedia.org/wiki/Linking...

Oh, hey! A satisfactory answer?? Thanks!

Always understood it to mean “I’m getting there, but not yet”. For example, I can say I’m pushing 40, because I’m 38.

Definitely nearly.

Approaching that age I would say

a) nearly that age, as in very close to. 59 is pushing 60, 58 is not.

Nearly, but it's often used ironically: "Now that I'm pushing fifty--" "You're fifty-three!" "I didn't say which side I was pushing from!"