My 13Pro only displays the clock for ~30 seconds before turning the screen off (when I do the turn it sideways while charging thing, "Standby mode" I think its called?)

I think the always on screen is a 14+ feature.

Oh yeah I definitely forgot about that.

Actually probably not. Does it have the same size as the 14 Pro ? If it's the same you can use it with no issues. All the orange photos you see are from an 14 Pro, so it should work nicely.

I looked into it and supposedly 14 Pro is 0.8 mm taller and 0.2 mm thicker.

You should be fine, go for it :)