I run faster than him – you cannot say it like this, as it is grammatically incorrect. The sentence must be “I run faster than he”. This is the short version of the sentences following but it is very colloquial, and you would just not say it, use the long version below, that’s at least a correct sentence.
I run faster than he does - correct
I run faster than what he does --- not correct. He runs, he does not do a thing. You cannot compare two different actions. What always refers to something mentioned before. And in this case the does won’t be an auxiliary verb, therefore you have a weird sentence with no meaning. The pronoun I always is written as a capital letter.
this place is what we were looking for...can’t it be written as...this place is the place we were looking for. Exactly, you could say that too, the difference is that “what” in this context refers to “this place”! This was mentioned before, and the what only prevents you from repeating.
OR this size is bigger that (I guess you misspelled it, you did it right everywhere else, the correct word would be than) what i am wearing...could be...the size is bigger than the size I am wearing
In all your examples the “what” refers to the object of the sentence, replacing the repetition, making the sentence smoother