Official government website from which you may apply.
Do speak with the Royal Thai consulate in your area. If you're in California, there's one in LA.
Your wife can also gain insight from speaking to other Thai spouses in LA; she/you may be aware of the Thai Chamber of Commerce in LA and, of course, little Bangkok. 😉
Do forget about the elite visa entirely. It's a waste of money for you.
You're old enough to get a non-immigrant O based on retirement or marriage... though is your marriage registered in your wife's local Amphur?
Read up on the government website please; search this forum a bit more; this issue has been covered many times, mate.
As for travel in and out, you'll need a "multiple re entry permit; which is completely separate from any visa.
I reckon things will also depend on your American laws concerning your wife wanting to keep her green card or not...
You do realize you'll not only have to attend but also make progress?
Are you aware you'll also be interviewed in Thai by the immigration officers to confirm such proficiency progress each quarter, as they review your request to extend?
Finally, most "good" schools are booked up for the intro/basic levels already.
The top three schools listed here in alphabetical order.