[Snake]

Ice Point Guard persists, even if the wielder of the buff swaps classes. Intentional, to give everyone a free guard and +10 DEF/RES for not using Verglas?

[Neus]

These should end if you start a battle without Ice Point Guard equipped. Greaves has also been changed to behave similarly.

Any other class buffs that last through multiple battles that actually make an impact like this one?

[Snake]

Any other class buffs that last through multiple battles that actually make an impact like this one?

Not a class buff, but Weapon Charge (from Excel weapons) behaves the same. They stick with you post-battle.

Lantern Bearer's last lantern aura used also sticks post battle, to the point it triggers Raging Flame without the aura invoked on the next battle. (Making the current aura 'invisible', but this passive still effective. Another thing that I'd not dislike having. Invisible LB auras behaving like Fear, icon-wise. [Fading in, then fading out after a fixed delay.])

Demon Hunter stances too. (Though, not necessary to be changed in my opinion.)