No, there's no restriction on flights they can fly. Just pick it out of the schedule, out of any airline, and fly it. Any restrictions would be upto the individual admin/airline to work out as policy, and accept/reject PIREPS in accordance to that.
On registration they can chose which airline they belong to, but it has little effect other than aesthetics really. I guess you can think of it the pilot belonging to a "parent company" with all the airlines you setup as divisions under that company. The separate airlines are just another means of organization