Is he the one at the top? That is a palamino.
"He can't be TWH.. Or at least not fully... Because he's got appy patterning, and white-walled eyes." this is not true. A TWH can have spots like yours does and the whites of eyes really have nothing to do with breed. With no more than the few pictures I scrolled quickly through I would say the horse is flea bitten. If he is registered or you know one or both of his parents I can help you further.
But here are the basic genes:
W- white is dominant. White hair has pink skin. Your horse is white.
E- can the horse PRODUCE
A- is the black hair restricted
to the points?
TO- tobiano, yours is not so it is toto
Cr- the creme gene, dilutes coat colors
D- dun gene. Produces darker points, dorsal sttripe, arrow over withers, striping around knees, etc
Here is a fun website to explore- Horse Color Genetics: An Introduction
Like I said- if you know anything about a brother or sister etc or his parents I can give you a better idea of genetically what he is