Birch sap does contain sugar but not a lot. You have to boil 100L of sap down to get 1L of birch syrup (while for maple syrup you only need to boil 60L down to get 1L).
Fermenting the sap directly you’re not going to get much of anything, probably less than 0.5% abv if you don’t add any other sugars. I know you said you are going to add sugar, but I do wonder if you’ll be tasting much flavor from the sap at all if you don’t concentrate the sap.
For super cheap pseudo temp control, you could build a swamp cooler, basically put your carboy into a tub of water with a t-shirt or some cloth around it to soak water up around the entire thing and then blow a fan on it to create evaporative cooling.
Another option would be to use kveik yeast and then you don’t really need to worry about ferment temp (other than maybe being too cold).
That yeast isn’t dead, it’s just gone dormant as there’s no food (sugar) left in the bottle for it to eat. You could easily build up more yeast from the stuff in the bottom of one of these bottles.
There are certainly better places to harvest yeast but that doesn’t have anything to do with whether or not the yeast in the bottom of a naturally carbonated bottle of beer is alive or dead. It’s not dead, just dormant.