Yes, you either need to show up in person with your kid and the mother (along with your passport) to sign your permission, or you have to send in a permission letter along with a copy of your passport.
See the following infographic by the Consular Department (in Thai):
Yes, all passport holders can apply for a visa. Some passports (not including UK) will take longer to approve if you do not have proof of residence in Cambodia.
It's up to the Thai Embassy you're applying at, but in most places if you're married under Thai law (and thus have a Thai marriage certificate) then there's no need to translate/legalize. But check with the Embassy you're applying with to be 100% sure