Oh cool, that's right up my alley, though it sounds like you know far more than I do (good for me, sucks for you). Hyperreals, I just looked them up, seems to be very similar to Surreals, just not a proper class. I'll have to look into them more someday soon.
Covariant notation seems a bit like reverse Polish notation for formulas. I only remember this because I had some graphing calculator back in the day that used it, or at least something similar to it, for a while. Was honestly never easy to make the adjustment, and Texas Instruments won the battle for graphing calculators eventually so I didn't keep that one around very long.
Good stuff. Nothing to ask about jumps to mind at the moment, but glad to meet another math enjoyer on here :)