I have had more than 800k with Bangkok Bank without any issues for 22 years. It’s perfectly safe and easy to use. But you won’t get a lot of interests on your savings account.
As mentioned above, each branch will have its own rules. Statistically the easiest banks to deal with are Bangkok Bank and Kasikorn, and branches in areas with many farangs. Usually you need your passport and a residence certificate which you can get from the immigration, or a yellow residence book if you have one. Better go with your wife.
You cannot get married in Thailand if you are already married somewhere else. You can register your foreign marriage at your local Amphur, that was the requirement until now, but it seems it is changing.