All true with one exception. And this is a pet peeve of mine.
Normal climbing has no DC.
A Dwarf in full armor with babies strapped to him and a dwarf with no gear and pirate clothes can both climb a tree with no check. Especially since the the babies likely don't reach the dwarf's encumbrance.
A dwarf with a single baby... that might require a check, but considering it was no check before, should it be a difficult check? I don't think so.
But, the problem is that people seem to think that every attempt to climb has a check. Which is frustrating, because then every low strength character trying to scale a tree or climb a rope is getting penalized when they should not be. Which then makes mildly difficult climbs nearly impossible.
But, per RAW, it takes a fairly intense situation to even require a check for a character to climb.