I want him to run.
"To" is not a preposition here but a subordinator that serves as a marker of to- infinitival clauses.
"Want" is a catenative verb and this is a catenative construction where the subordinate infinitival clause "to run" is catenative complement of "want". "Him" is the direct object of "want" and the understood (semantic) subject of "run".
"Him" is called a 'raised' object because the verb it relates to syntactically is higher in the constituent structure than the one it relates to semantically.
Incidentally, I would say that in your other two examples "next to him" is a PP modifying "girl", and "to the store" is a PP serving as complement of "went".