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Question

Are dhampirs affected by the cold since they are considered undead? Also does ice deal any negatives to the spider climb ability.

Answer 1

You're looking for a level of detail the game generally doesn't cover

That's kinda the short of it. No part of the lineage's text references cold, so they are affected as normal by extreme cold and cold damage, and the spider climb trait has no clause related to cold damage (or cold weather), nor about ignoring non-verticality aspects of terrain so icy surfaces have the same difficulty as when walking on it (though possibly adding a threat of falling if your DM feels like it). Player options generally haven't got that, but for an example you can look at the regeneration trait of trolls (or vampires) which turns off from fire (or radiant) damage.

When it comes to cold weather, the game doesn't really describe that other than extreme cold (DMG p. 110), meaning for normal cold you are free to roleplay that in any fashion you'd like, equally free to say your dhampir character doesn't care about the cold, or (since they're no longer warm bloodied) actually find their joints struggling in cold temperatures (the latter might be more apt if you've been failing a bunch of checks this session).

Answer 2

Dhampir are affected by cold. Ice does impact spider climb if the ice is slippery.

First, there's no special package of rules for being undead. Most have resistance to poison, but Vampires are exceptions to that pattern. The Dhampir will only have the properties listed in the source materials. A DM might rule that being cold doesn't bother them for a role playing perspective because of their undead nature, but that doesn't affect game mechanics.

Second, the spider climb ability does not negate difficult terrain. Slippery Ice is listed as a possible difficult terrain on DMG 110, and suggests that there should be a save against slipping. So it'll depend on whether the ice on the wall is determined to be slippery.