What is the ruling on posting pictures of drawn characters (anime) on social media, such as using them as a profile picture or in a post, and is it permissible to cover the objectionable parts of a nude picture and then post it?

There is no objection to using anime images on social media or elsewhere, as long as they are free from Sharia-prohibited elements. However, if the image contains something objectionable, such as nudity, then it must be avoided, even if the intention is to cover the nude part, in order to avoid looking at the nude image in the first place.