I would like to assign field(s) in the database with a value that identifies which user created/owns the contact. Is there a way I can either assign a constant value (integer, username, etc.) to a field? Or manipulate the INSERT/UPDATE sql statements directly to add custom functionality?

I know this could be done with a trigger on the table but I would rather not add this globally (only for this integration).
You can add a mandatory key (assign dialog->mandatory keys) with some sql statement that is
generating user name/identification, but this will work only during insert(not update).

the statement depends on your database but it will be something like:
select GETUSER()