you need to go to local immigration and asked for certificate of residency for purpose of car purchase. In Udon Thani, it's THB 500 for each certificate.
Your rental agreement is OK if it shows copy of the Condo title deed and owner's Thai ID card. If you use this rental agreement to get certificate of residency more than twice (eg open bank account etc), they will ask you to come back with Yellow book in future. That's what the IO said to me in UTH.
Don't leave it in the bank. Buy physical gold. When we first arrived in Thailand more than 2 yrs ago, it was THB 28,000/bhat gold, today it's THB 46,600/bhat gold. 1 Bhat gold is 15.2g.
Definitely put your mom's name on the lease as its easier for her to request for any certificate of residency from immigration in future when need arises. Also request landlord for copy of their ID card and their Blue Book. Immigration in Udon Thani insists on seeing these documentation on top of the lease agreement.
I got a True broadband and a basic cable TV package and a free mobile SIM came with it (Udon Thani Central Plaza) . I pay THB 680+/month. You will need your condo lease and passport.