Check out sites like renthub and hipflat, can then contact agents through there for places you like. Alternatively, use the site to find places you like and then go to the condos themselves and speak to the juristic office to see what's available.
it will most likely be the airline who will check at check in more than immigration once landed. I don't know if they check to see if your ticket us legit or not at check in or just a quick scan to see you're leaving Thailand within 60 days.
don't worry about how busy it is. I live in Bangkok and moved here when my youngest was 2. I used to push him around in a pram, no issues. As someone mentioned elsewhere, make sure the pram has good wheels as pavements aren't great here.
you don't have a degree so you can't legally work here. You also don't have enough experience to meet the requirements with the exception of teaching English.