It can take at least 6 months after your child is born to obtain custody rights because you need to go through the Courts. Your best bet is using a 'border run' agency.
Not strange at all. Been stopped at road blocks a few times, never been asked for an IDP only a driving license. It's not actually a requirement to have one.
Just get the Non Imm O before you arrive. You shouldn't have problems opening a bank account. You wont need onward travel to board your flight or enter Thailand, and you won't have the extra trip to an Immigration Office to apply for the Non Imm O in Country