Yes.
Thats the mundane way of recovering wounds from battle. Also called Natural Healing. Any character can recover wounds by resting for 8 hours, limited to once per day, regardless of how much he rests or how much aid he recieves.
The Heal skill simply allows a wounded character to recover faster.
You will notice that Treat Deadly Wounds says the character must recieve treatment no more than once per day, and no longer than 24 hours after being wounded. Which specifically prevents that type of treatment from being used more than once.
But for Long-term Care there is no restriction, other than spend 8 hours treating the patient(s). That is actually the slowest form of healing available.
Also keep in mind that the character recovering for 8 hours will recover only his double healing rate (2 hp per level) from your Long-Term Care, and not the quadruple (4 hp per level), as he requires a full day of complete rest.
For a complete rest, the character must make no stressing activity for the day if you want to recover. What is a stressing activity is left to the GM, but most assume that any skill check, combat, travel or spellcasting counts as stressing.
You can even Provide Long-Term Care with your downtime. Or while Travelling (which is a common use of the skill), as that is considered light activity for the healer.
Giving long-term care counts as light activity for the healer.
Which means you could even treat someone while recieving long-term care yourself. You simply cannot treat yourself this way, or recieve quadruple the healing as that requires complete rest.
A Mobile Hospital can be used to double the amount healed aswell.
Forgo isn't the best word choice, Assurance works as rolling
The clear intent of Assurance is that you no longer have to actually roll a die, but the results should be adjudicated as though you had. Anything that references "rolling a success" or similar is functionally the same as "getting a success" and Assurance affects either.
Normally I would take words at their face value, but we have to remember that Paizo has a number of writers and has adopted a stance that things don't have to be written exactly the same to mean the same thing (vis-à-vis "rolling" vs "getting"). And we know that Assurance is still a type of roll because it's a Fortune effect.
A fortune effect beneficially alters how you roll your dice. You can never have more than one fortune effect alter a single roll. If multiple fortune effects would apply, you have to pick which to use. If a fortune effect and a misfortune effect would apply to the same roll, the two cancel each other out, and you roll normally.
A fun side effect of this is that Assurance can be negated with a Misfortune effect and vice versa; if you have a Misfortune effect on you, you can use Assurance to roll normally.
Because Assurance is still a type of roll (even if there is no physical/digital rolling of a die), it qualifies for any text referencing rolling a [result] (assuming 10+proficiency is enough to succeed).
Best Answer
You can increase the DC based on Nature proficiency
Alright, the rules are mostly silent on this, but there is good evidence that you should perform Treat Wounds in a way that is essentially replacing the word "Medicine" with "Nature".
The writing could just be different authors, but I think the intent is pretty clear; the new skill can be used like the skill it's replacing. I think this is true of all of these abilities, but Natural Medicine is the most clear that it should completely replace its original skill for the given Activiity.
*Natural Medicine is already of dubious utility given the importance of other Medicine Feats (which it does not allow you to qualify for) to the Treat Wounds activity, unless you are taking Herbalist Dedication.