What are the immigration office requirements and timelines upon arriving in Bangkok with a Non-O visa?

Within 24 hours after arrival, Immigration office for filling the TM30 form. Wherever you might be staying they have to fill it with Immigration (landlord, your wife, hotel, AirBnB, whoever owns the premises). NOT you, THEY must do it.

Edit: in the sticks, they would not even know what you are talking about. Whover in that place owns the "Blue Book" has to go Immigration with you, the next day, with their Thai ID and sign you in for the TM30.

TM30 is needed if you would come to Thai Immigration for any service.

If not, just ignore it.

NO visa? Non Imm O visa? Timescale to do what?

I assume you mean a Non-O visa? That gets you stamped in for 90 days and at most offices you go and get an extension of stay when you have 30 days or less left on your entry stamp, some offices are 45 days. But bear in mind that you also require funds in a Thai bank account for 2 full calendar months before you apply for that extension.