Wow, all those condos sound really nice!
Once I started building I found that it was pretty stable how I put my cubes together, so I just left the outer frame as all cube and zipties. However mine if only 3 cubes high, I can see how going four high you’d need the extra stability. I liked the idea of utilizing the baseboard in the stability of the levels, but this apartment’s baseboard is basically rubber things and I really didn’t want to have the condo depending on the walls (if that makes sense?). I was also working with limited supplies and tools because I don’t have the same access to my parent’s shop that I had at my old place. However, I think my mom taught me pretty well in the art of being handy because it all came together quite well.
So here are the finished picture:

I ended up finding slightly bigger square dowels to support my levels and screwed them to the platforms to create stability. I might add a ramp or two eventually, but Peach has always been a good jumper/climber. My frame was pretty stable without extra wood, so the other frame is just cubes and zipties. I used a sheet of vinyl flooring (that funny enough matches my flooring) and ziptied it on two sides of the condo. The back as some excess that is just pressed between the condo and wall. My platforms are wood planks with peel and stick vinyl tiles on them then covered with fleece. The door is really complex and I need to eventually make it more secure, but the whole middle two front cubes open all the way up.
There is a towel in there right now (but he’s pretty good about leaving towels alone) because he proceeded to “defleece” that platform and then drag it to bury his water, so it’s drying in my bathroom.
I’ll move it back after cleaning it. I also moved the hammock he hated in there to see if he will try it again, he’s unsure but definitely likes it more there than how I had it in the other condo, so we will see.
He seems happy with it. He’s pouty right now because he’s had a lot of excitement this last week and I think it’s catching up with him, but he’s my good boy and I know once he realizes all the nooks and crannies I’ve situated for him, he’ll be very happy 
Thank you again, everyone, for your advice and I hope my condo maybe gives others some ideas and inspiration.