What are the requirements for switching a tourist visa to a work visa and obtaining a criminal background check from abroad for Thailand?

So, the visa bit was more or less answered: anyone *can* switch from tourist (or visa exempt stamp) to a non-b but it's really up to the employer and whether they're willing to do the in-country switch -- most schools do prefer to send you out-of-country on your own dime -- noting that now it's done online and this sometimes can create problems with timing but that's between the employee and employer.... It is totally reasonable to expect to have to go out for the non-b visa and return though, nothing particularly crazy about that

...the second part, about police check, don't overthink it

I'm Canadian and I had a police check done online and sent to me, should be very easy and not more than maybe 50 bucks. This should work for your first background check as you've been living out of thailand it can't be from Thailand (it needs to be from your home country but it doesn't need to be an "FBI" or overly complicated/official thing (online is more than sufficient to say you have no criminal record)

After 1 year of teaching, for any consecutive years after that, the police check will come from Thailand (the province where you end up living and your husband's school will tell him how and where, and it will need to be done every year for each year he renews his work permit)

Standard practice in Chiang Rai is to get cheap local police background check which takes like two weeks, then travel to Laos to get teaching visa. Normally the school does all the work for this and communicates with you.

Whether or not you can convert from tourist to non-B inside of Thailand is completely up to your employer. It requires a massive amount of paperwork to do an in-country conversion, and many employers cannot or will not go through the trouble. The vast majority of employers require you to get the non-B from a Thai embassy outside of Thailand, which requires significantly less paperwork.

The background requirements are whatever the place requiring it says they are. Is it the embassy requiring it? What does that specific embassy say? Is it an immigration office requiring it? What does that specific immigration office say? Etc.