No, if you mixed in enough cider vinegar (50% - and vodka is better as a preservative) then you should have no problem taking a small bottle along while you travel so long as you keep it from getting too hot and keep it out of sunlight. Like most organic things, it does keep better and longer in the fridge.
I have some now that I kept in a dark closet and did not preserve well enough that developed a bit of mold on top. I just strained it, boiled it again for a bit and refiltered it and added some more preservative. No mold since then and it should be fine to use, but since I have plenty of the commercial supplemental form, I am probably going to boil down the homemade extract to use in a skin cream.
No, you mix the extract 50/50 with either vodka or apple cider vinegar. I think vodka is a better preservative.
You don't have to use the leaves the day they were picked, but the sooner you use them the better. Certainly do not let them sit around and begin to spoil.