What we'd suggest taking and doing, based on what's worked (and what hasn't).
Most tea houses are happy to have a dog to stay, but "happy" sometimes needs a bit of coaxing — some owners assume a dog means a mess to clean up, and a local guide vouching for you goes a long way. It also helps that not everyone in the mountains speaks English; having someone who can explain what you're asking for, and why, smooths over a lot of otherwise awkward moments.
Make sure your dog's vaccinations — rabies especially — are properly up to date well ahead of travel, not sorted the week before you fly. It's worth starting tick and flea treatment in advance as well, rather than waiting until you're already on the trail and it's too late to do much good.
If your dog likes to sleep on the bed, bring a sheet to put down first. It's a small thing, but it's the difference between a tea house owner not minding and a tea house owner minding a lot.
Not every tea house serves meat, but where they do, most are willing to boil up some chicken for the dogs. Where they don't, mine get by on noodle soup with egg, tins of tuna, or dhal bhat if nothing else is going. If your dog will only eat proper dog food, plan around that rather than hoping — I always carry at least six sachets, just in case a village turns out to have nothing they'll touch.
If your dog is reactive around other dogs, I'd hold off on a trek like this until you've worked through that training — the trail isn't the place to find out. The same goes the other way: plenty of people are genuinely afraid of dogs, however friendly yours is. Even a dog that adores people should stay on a lead until you've got a read on whoever you're meeting.
A little goes a long way. "Tokdaina" (doesn't bite) is the one that gets the most use — said with a smile, it reassures people faster than any amount of explaining in English ever will.
Beyond the paw care covered on the Dangers page, it's worth carrying basic medicine for your dog and knowing, before you set off, what you'd actually do if they got properly injured or ill — where the nearest help is, and how you'd get them there. Hoping it won't happen isn't a plan.