What is the highest temperature human skin can withstand with no risk of burning if applied over a 1 sq. area?
The reason I am asking this question is because I have found that simply applying a hot wet rag compress for less than 5 minutes in the very early stages of a staph infection will stop it in its tracks instantly. The key is identifying the very early stages. I could never understand why my Dr. never took it seriously when I mentioned it to him but it finally hit me that a Dr. would have no experience dealing with very early stages. I happen to be an expert on this for reasons I will explain if asked. But it behaves very predictably. and I would bet that most of us have experienced this.
The first 4 hours just an occasional itchy spot, next 4 hours, the itch becomes more frequent with twinges of pain and a hard spot starts to develop under the skin. ( This is the stage where it needs to be caught) Next stage will have a couple of hours of fairy intense burning and then redness with start to set in with tenderness and soreness. From this point it can spread but usually just results in a nasty boil. I have had it work in the intense burning stage but it took a bit longer and I had to get the water hotter than my sink.
I am going to take a wild guess at between 115 and 120 degrees would be the highest safe temp for periods up to 5 minutes.
I could not say. A first degree burn is essentially like a sunburn and a second degree burn has blisters, etc. - so it depends what you mean. It might make sense to make it cool enough not to leave a lasting mark. I doubt the benefits of going hotter would be very dramatic and your immune system will probably work better if the basic elements are still fully intact. Rather than go much hotter I would consider adding antibiotic, which often works well and especially at early stages. The hot tap water I use for shaving is a good temperature, though I have never measured it.
I was thinking a temperature where the redness would be gone within 10 min or so.
If it helps any, my tankless water heater comes with a cap of 125F on the temperature because anything over that is likely to cause a scald. I like really hot showers and recently tested what I find to be a comfortable temperature and discovered I prefer my showers in the 112-117F range but most people think that’s quite uncomfortable, although not damaging to their skin.
Also, one very common cause of recurring staph infections is having been spider bit at one time, then the staph bacteria likes to set up shop in your nose, leading to repeated infections on your own and other’s skin that might come in contact with your mucus. Putting a nice wipe of neosporin ointment up your nose three times a day for six weeks is usually enough to kill off the colonies and reduce your recurrent infections.
I’ve found 140°F pieces of metal are touchable, hot, but I don’t think will cause a burn. It’s a quick test for overheating machinery if you don’t have a thermometer handy. 160°F is too hot and will cause a 1st degree burn in a few seconds. Hot water will be different, it transfers heat very fast over a large area.
Many years ago, I spent some time working for a neurobiology researcher who was studying pain. I would design different machines for him to test various pain thresholds. So technically I designed torture equipment. 
When testing the heat pain threshold, he had me set the hot temperature safety cutoff at 50 deg C. If the temperature of the probe ever got hotter than that, the safety circuit would shut the machine off, regardless of what the computer was telling it to do.
Above 50 deg C (which by pure chance is a conveniently round number), the skin starts to burn, and you get damage to both the skin and the nerve endings.
For reference, there are two “hot” pain receptors, called TRPV1 and TRPV2. TRPV1 activates first, typically around 41 to 43 deg C (IIRC). TRPV2 activates at a slightly higher temperature, typically around 44 to 46 deg C. Capsaicin (a chemical found in hot peppers and other “hot” things) activates TRPV1, which is why hot peppers are “hot”.
Cooking sous vide with an immersion circulator gives some insight into pain thresholds. I can’t remember the exact number but i think about 130° F was too hot for comfort if more than a few seconds but tolerable for a quick hand dip. And today, I learned my phone’s keyboard has ° as well as these °•○°●. Bubbles!
That’s about 122° F which jives with @SmartAleq’s tankless safety limit.
My father was a doctor, and he frequently recommended hot compresses for minor infections. He claimed that the hot water increased blood flow to the area and alerted your immune system (like a localized fever) and it wasn’t that the heat killed the bacteria, but that your white cells did so.
I agree with @Dr_Paprika that adding a little OTC antibiotic ointment helps, too.
But I’ve had excellent luck treating minor skin infections with those two aids.
Because one wishes to avoid burns while bathing, there are probably numerous web sources suggesting a limit on your water heater. I’d look for those and dial it down by a comfortable safety margin. However, if your question is more than just asking (you plan to do it), again I would suggest the benefits of a few degrees are likely modest. If the skin does not stay red there is no burn.
If this is a chronic problem where the boils are in and under the upper arm, this can be hard to treat. There are medicines targeting the immune system, but I have no experience using those for treatment. It is not my business. You have my sympathy if this is so, and if not than please ignore this paragraph.
About 25 years ago I owned a sharpei dog. They have very prickly fur. The dog used to like to sleep right up against me at night. I ended up with close to 30 major boils in about a 3 month period. The Dr. was calling them spider bites but I always felt they were staph infections. My wife at that time was a trauma nurse in a county hospital and I think she may have been the vector for the germs and the dogs prickly hair was kind of injecting these same germs slightly under my skin. That was when I discovered the hot water compress trick. I had become very aware of when a boil was about to develop. It was 100% effective and the results were always immediate. This morning I felt one coming on at my hip. Put the hot compress on actually for less than 2 minutes and it was immediately resolved.
The wife would come home and I would give her a hug before she changed her uniform and I seldom was wearing a shirt. I might lay back down to watch TV and the dog would lay right beside me. I am pretty sure that was what was happening.