At the moment, only have an opinion on the first one. I spent considerable time tweaking hit point recovery for my current campaign, both before it started and then after some initial play. Here are some thoughts that may not exactly fit what you want, but may help:
A. You can set the recovery time for short/long rests to whatever works for the feel you want. Whether that is 1/week, 1/day, or something else, it can all work, if close enough to the desired feel.
B. The key to me is some variation of another option in the DMG, where hit points do not automatically come back with rest. Instead, Hit Dice must be used for recovery, and then most of your recovery rules deal with getting Hit Dice back, not hit points. Regardless of the rate chosen, this point will allow characters to be worn down slowly at times, more quickly in others.
C. I had several (semi-complicated) rules for the recovery of hit dice. After some play, I threw them all out for a simpler version. The default is half the hit dice recovered on a long rest (round up). Then I modified that by GM adjudication if the conditions were notably good or bad for recovery. Instead of set rules, I gave the players some examples of what could contribute to a better or worse rate. Those include hot food or not, the quality of the shelter compared to the weather, etc. Sleeping under the stars when it's pleasant out in the fall? No problem. Try the same thing in winter, better find a cave or construct a shelter instead, and arrange for a fire.
D. Working the exhaustion rules into the mix more can cover any holes in the above. There are several things you can do, as it fits what you want, such as having death saves (failed or even attempted) cause exhaustion, making exhaustion need to heal before hit dice can, and so on). I wanted exhaustion more tired to the worry of directly dying than hit die recovery, so I did the first one. But other approaches can work too.
If you think of rests, hit die recovery, and exhaustion as your three tools, it is not too difficult to allocate different pieces of the natural game play to each one, in a way that will allow the players to simply try to get good conditions for themselves, and let you handle the mechanics. That should help immersion on that front. Hit dice available then become the clear proxy for how much natural healing can happen.