Every friendship is different. I have some friends that I only see a couple times a year, but we text every day. Inviting him out once a month isn't excessive at all. Since you're still having other positive interactions, I would just continue inviting him to low pressure things.

Inviting him to group events might be good, too. For example, "I'm going to be hanging out at this place at this time with these people if you want to join."

Maybe he's just busy during lunches 🤷‍♂️ I'd try a different activity and see if you get the same response, but I wouldn't be trying too hard

No more than 2. He doesn't want to hang out with you. Move on.

Trusting your gut makes sense. It sounds to me like he's got a medium level of interest, but isn't in a hurry to hang out terribly often. Part of it could be that, since he already lives in the area, he's just going to prioritize existing family and close friends over a possible new one.

Making friends in a new place can be frustrating. Just find various kinds of meetups (sports, volunteering, hobbies, clubs, etc) that interest you and eventually you'll pick up a couple of friends. You have to go in with no expectations for any particular person, but put out invites and see who ends up wanting to hang.

Good luck, keep trying! It took me a year to build up a base of friends when I moved fairly recently.