Waiting for bro rotikosong to come in, he's like the expert of these northern mountain places. Hey where the hell are you?
Pai seh pai seh. Been crazy busy traveling for work so no time to come in here.
Most of the questions already answered thanks to the rich experience of others here. I try my hand at some answers:
>> How and how much to cross into Tachilek?
IIRC 400B or a crisp USD10 note. This you pay to Myanmar immigration and you get a one-day pass. They keep your passport and you get a paper receipt.
You can also ask them to stamp their entry stamp on a loose blank sheet of paper. In the past, this was valuable as Myanmar was a pariah state. These days probably not done anymore because Myanmar is mainstream.
Tachilek tourism is mostly wat (you can see their famous Schwedagon stupa [not the Yangon one but same name] from Mae Sai) visits, casino, golf and their street market where most things are super fake. The type of place you go once and say ok, I'm not coming here again.
>> Laos and Myanmar in 1 day
Crossing in Laos is easy from Chiang Saen which is the next big town over from Mae Sai to the east. You basically cross the Mae Khong river to the other side and you're in Laos. Chiang saen is perhaps 1 hour drive away from Mae Sai. People (local tourists) go over to Laos to shop and again visit some wat. No passport is necessary, just a boat crossing really.
>> CM - what to do?
Already answered by YY who as always gives excellent lay of land. First visit to north, I'd recommend CM because there is nothing to see in CR except for the White Temple (Wat Rong Khun) and really nothing to see in Mae Sai at all. Those who go to Mae Sai are there for other reasons besides pure tourism.
CM on the hand is fascinating for the first-timer. Here you have an inner city surrounded like a fort by a moat, with bridges and gates into the city. You either stay at Nimman or perhaps more more popular would be the Night Bazaar area. Lots of temples, lots of shopping and nature and outdoor activities like hill tribe visits, white water rafting, elephant rides, ziplining etc.
One popular activity, especially for Thais, is a stay in a wat. They go for 2 days to a week and stay and pray, and live in dorms within wat. No phones, no electronics, just meditating and chanting and listening to sermons.
If you go to CM be sure to rent a motorbike (if you can ride one) to get around.