Is there a genitive form of the word this in English?
- The color of it -> Its color
- The color of this -> ?
This, that, etc. are demonstrative pronouns and thus don't have genitive forms. The genitive case would be attached to the noun that you are demonstrating, e.g. "This thing's colour."
That isn't to say it wouldn't make sense to have genitive forms of demonstrative pronouns - their usage would be fairly niche - just they're not currently a feature of the English language.
No. In The Syntactic Phenomena of English, McCawley uses this fact to construct examples illustrating his proposal that the purpose of syntactic transformations is to fit logical forms into the surface requirements of English, one of which is that all the words must be permitted by the English morphological system. If you find yourself wanting to say "this's", you just have to find a different way to say it.