Do I need to stay in Thailand for the 3 months after applying for a retirement O visa before applying for the one-year extension?

Here's how that works in country

You apply for the 90 day Non-O visa for 2000baht at the immigration office

It will go under review for between 2 and 3 weeks (depending on the office). You need to have the time on your entry stamp to be here during that period and during this time you can't leave the country

Then you go back on the date they put on your receipt when you applied (in bangkok 2 weeks later), you get the Non-O visa and a new 90 day stamp inked in

At this point IF you need to leave the country you buy a re-entry permit. It will ONLY be good for the 90 day stamp you have so you need to come back before that date so you can apply for the year extension (which you can do when you have 30 days or less left on your 90 day stamp and your funds have been in the bank for 2 months)

So the answer is yes, AFTER you get the 90 day Non-O visa you can exit/re-enter during those 90 days IF you buy a re-entry permit