The source for Kiddushin is the Talmud Kiddush 65b. This is not a universal opinion, (even in Halacha, see SA EH 42:2) as some allow for one witness if both agree - at least to not ignore the Kiddushin.
The distinction between money (since the source for requiring two witnesses is money) is described in the Talmud - because it obligates third parties. That means that his relatives can't marry her anymore and vice versa (Rashi there).
It is one of the most famous Yeshivishe discussions that the distinction here is that money the witnesses are Eidei Birrur - they clarify the reality, so that if we know via other means, such as the parties admitting, then it is the same thing. But when it comes to Kiddushin they are Eidei Kiyum - they make the event happen. Without them, nothing actually took place.
A kesuba needs to be signed because without witnesses there is no property lien and that would limit her options to collect, and the Chachamim require her to have a Kesuba that satisfies her worries (Beis Yosef EH 61:1 quoting the Ran, who also quotes and disagrees with the Rambam and Itur as it being just the Rabbis requirement of the Kesuba unconditionally, which would also seem to be the implication of the Tur there as understood by the Bach).
Chuppa/Nissuin requires witnesses the same way Kiddushin does, under the same circumstances (because Kiddushin establishes the principle that witnesses are required to create the fact of the matter, and this applies to Nissuin as well - it doesn't happen without witnesses). So the way it is required for Kiddushin, it is required for Nissuin. I don't have a great on point source for that, but I found it in many discussions of the issue as a given. (In Yabia Omer (I don't have it to check exactly where) he tries to get out of that because that would mean that Sefardim aren't following the Mechaber's opinion and quotes some Rishonim who disagree, but he agrees that having witnesses has a good source).