Regarding the border run to Laos across the Mekong (not a small river) from Sop Ruak (not close by to Mae Sai), I posted this back in February and presumably it is still accurate.
Go to the immigration office in Sop Ruak. They will stamp you out and they also ask if you have at least 10,000 baht in cash and an outbound air ticket. (On arrival, they did not ask to see that).
After getting stamped out walk out the back door which leads to the boat dock. The boat across to Lao costs 70 baht if there are at least three passengers and they run on demand.
Once on the other side, you disembark and walk up to Lao immigration. The first window checks passports and hands you an arrival form, which you complete, then hand it to the next window and then pay at another window. The visa cost varies by country - for Americans it was 1,500 baht. There is a small duty free shop inside the Lao arrivals hall just after you enter with decent prices on spirits and wine. If you have USD, use those as they were doing 37 baht/dollar for purchases.
You can then go to departures, get stamped out and return to Thailand to get stamped in. The boat back to Thailand costs 150 baht.
Once back on the Thai side, you just need to go through Thai arrivals and you're stamped back in. The immigration officer advised and confirmed that they will only allow this twice each year.
All together this will cost you 1750 baht which is more than the Mae Sai crossing would cost you, but it’s there if you need it while Mae Sai remains closed.