1- Imagine a contract is like a cover, a blanket. You remain under your blanket so that you keep yourself warm, because it is cold out there ! I think about it in this manner, i.e. figuratively.
2- As i understood when you say "on a contract to", what comes after "to" is an infinitive.
3- As I understood "on contract" functions as an adverb.
4- we can say "has a contract with". I am not sure if we can say "remain under contract with" ("with" in place of "to"). I guess it is like "I got married to Jane".