I'm assuming that things are the same as the last time I through checked (5 years ago).
If it's one ticket booked through, then the bags will also be checked through and you should clear at Samui.
If they're separate tickets, or the onward airline doesn't have an interline agreement, then you'll pick up your bags and do immigration in Bkk- then recheck your bags.
I won't comment on this individual case as I'm not the expert. But when someone says "it's fine", I want their name on official headed paper with an official stamp.
You don't pay for extensions at a thai embassy as there are NO thai embassies in thailand. You pay it at the local immigration office wherever you are staying.
I think I'm right in saying that (generally) in and out on same day is fine (though I think that a few land borders insist on at least 1 night spent out). I'll doubtless be corrected if I'm wrong!