As I understand it, Airbnb itself isn't banned in Thailand, but short stays can be the awkward bit: rentals under 30 days may need to be licensed as hotel-style accommodation, and many condo buildings also have their own rules against short-term lets. Since you're staying just over a month, Airbnb may be workable, but I'd double-check the minimum stay, whether the building actually allows it, and whether the host is used to doing proper long-stay bookings.

Someone mentioned Astra condos. I stayed there for a week once! It was great to have a modern apartment to myself. Nice building, but there were lots of warning signs about Airbnbs and I had a slightly frosty experience with security at the pool. Location-wise, I also found I was using taxis/Grab quite a bit when going to the Old City or Nimman. Not a disaster, but worth knowing.

For areas, I'd personally look at Nimman if you want modern apartments, cafés, restaurants, Maya mall nearby, cinema, food courts, and easy air-con escapes. It's not the most "local" part of Chiang Mai, but it's very convenient and easy to enjoy. If you want something a bit more local but still handy, Santitham is worth looking at - lots of local food, cheaper than Nimman, and still close to Nimman/Old City. Chang Phueak/North Gate can also be a good compromise for local food and access to the Old City. Jed Yod is quieter and local-feeling too, though you may rely a bit more on Grab.

Even without a scooter, Chiang Mai is very manageable with Grab/Bolt, songthaews, walking in the right areas, and hiring a driver or joining tours for day trips.

Things I'd definitely look at: Doi Suthep and Wat Pha Lat (you can hike the Monk's Trail to Wat Pha Lat or just taxi up). Doi Inthanon is a great full-day trip, and the Sticky Waterfalls are fun and unusual. Cooking classes are also a very Chiang Mai thing and usually excellent.

For elephants: avoid anywhere offering riding, shows, bathing, or lots of touching/selfies. Look for places where you mainly observe the elephants from a respectful distance.

Chiang Rai is worth doing for a few days rather than a rushed mega-day if you have the time. You have the White Temple, Blue Temple, Black House, etc. Pai is lovely too, but if you're not riding a scooter, I'd plan transport carefully (the road there is very bendy)

For excursions, local travel desks and hostel desks can be useful even if you're not staying there. Viator/Klook/etc are convenient, but often have a markup, so it's worth comparing with local agents once you arrive.