This is my situation, too. No run, they're just out on the property all day and they put themselves to bed in the coop at night and we secure the doors. I have seen this called "partial free-ranging." "Free-ranging" is no coop at all. The chickens roost at night in trees or wherever. With partial free-ranging, they are housed in a coop at night. I don't know what it would be called if you let them out of the run for a just couple of hours each day. Casual ranging, maybe?
Yes. A balanced commercial feed should be available to them, but they do eat less of it because they get more food on their own. I have feeders available, but honestly, when I open the doors in the morning, they run right by the feeders to get to the good stuff first. Later in the day, they'll wander over and peck at the feeder, but they do seem to prefer foraging. Keep in mind also that the breed of chickens I have, one of the breed traits are that they are hardy foragers so the experience you have with yours may vary.