Are They Playing?

Hi all! For a month now I've been doing a slow introduction between my nine-year-old resident cat, Luna, and an eleven-year-old foster cat, Max. I am hoping to keep Max who wasn't doing well at his previous foster mom's house but need Luna to be comfortable with him as I live in a small apartment. Max has been living in my bathroom with a fire grate between him and Luna and things have been going well: nose sniffing through the gate and no hissing. Recently, I started letting Max into the living area. The first time they were a bit wary of each other (lots of tail twitching and watching) and that session ended when Luna crouched behind some furniture then charged Max, doing the sideways shuffle with her back arched and tail puffed. There was no hissing, but I separated them anyway to be safe. 

Today, I let Max out again and I think  they've been playing. Max has been charging around and Luna will chase, then turn and run away (he only chased her a little; she does most of it and is the dominant cat). He will look at her, then turn and run, then crouch behind something. She did the sideways shuffle again at one point, then walked away and her fur went down. Again, no hissing, but Luna tends to puff her tail a little when she's running around. They never quite make contact. Most recently, they both laid down several feet away, looking at each other with tails twitching. Then Luna dropped her head to the floor in a full on sprawl and made a "bruup" meow. Max put his head down too. I'm hoping this is a sign that they are somewhat comfortable with each other? I don't have a lot of experience with cat-on-cat interactions, except when my former feral foster kitten was badgering Luna, who was hissing and sort of trying to ignore him... So I think it's going well but am not sure what to make of the back arched shuffling or some of their other shenanigans... Any thoughts would be appreciated.

It sounds like playing to me; but I would be cautiously optimistic at this point and continue to supervise their interactions closely for now.  

Great, thanks! Today they ate from the same cat grass container, with their faces really close. They were like that for a couple minutes so that makes me pretty happy...

Max also just went to the bathroom in Luna's litterbox, and I'm hoping this will help "mingle" their scents rather than put Luna off but I guess time will tell... 

Oh that sounds like good progress!!   Keep us updated please! ​

So, the last couple of days have been kind of mixed. Max will usually race around like a nut and Luna will watch. Not that she doesn't have her moments, but they're usually on her own rather than with him. Two days ago he went up to her and she put her ears to the side and seemed uncomfortable. They stared at each other and Max tried to get closer and Luna growled. Then I gently broke them up. A couple times, Max has continued to sniff Luna past when she's comfortable and she'll hiss or just look irritated. Then today they were both climbing around on the couch and cat tree, not together but definitely interacting in what seemed a playful, interested manner. Yesterday they got close enough to each other that Max raised a paw at Luna like he wanted to bat her. I think his intent was playful but she edged away before it could go further. Yet, as long as she's up on a chair, Luna will sleep pretty calmly in his presence.

My plan is to just keep exposing them to each other when I'm home (and keep them separate when I'm gone) and I guess keep monitoring. I'm slowly working on "catifying" my space a bit, though some of the structural changes to the living room will depend on how lucky I'll be at second hand stores...