Rolling initiative at the end of a rest and keeping it isn't a bad idea-- getting rid of the hard stop where everyone rolls initiative is nice, and 5e's Short Rest/Long Rest system means that you shouldn't be stuck in the same place for very long. Having an impartial way to adjudicate the order of noncombat options, like your description with the caravan, is excellent and intuitive. I don't know how much it'll actually smooth out the combat/noncombat transition-- "surprise round, you three get shot at, now everyone roll initiative" is pretty common in my experience, but it certainly can't hurt.

On the other hand, I don't think that small, numerous modifiers to initiative is a good idea. Particularly with the sort of modifiers you're talking about; those seem much more in line with a revised Exhaustion mechanic. It's a level of granularity that really doesn't mesh well with 5e-- to be honest, I don't think it really meshes well with anything. Switching initiative in combat is annoying, but at least everyone's in the rhythm at that point. Out of combat, it's way too easy to lose track of and get mixed up. I also don't see how that sort of fiddliness plays into your goals of increasing tension. The last thing you want is for a situation where initiative matters to pop up and everyone stops to calculate their modifiers and argue for bonuses.