A tumbler is a rock tumbler....some water, some aquarium gravel and a drop of dish soap plus time cleans up clad, brass items, all kinds of things pretty good.
For iron and a few other rusted or crusty targets electrolysis is usually the method most use.
A ton of vids on YouTube about this, lots of advice about set ups on most of the forums if you do a search.
Most use this process on objects that are cool like old gun parts, tools and whatnot, most don't bother on most iron that tends to be junk or junk like but when you are new almost everything seems like treasure.
Even if what you are trying to clean many wouldn't bother with like those nails it's your stuff, your choice, and you will have that electrolysis equipment ready to go for the next cool thing you find.
There will always be another one.
That permission stuff, most park departments have rules about metal detecting on their websites or post signs about it if you are not allowed.
If you don't see any indications laws or rules against it you are probably good to go.
Asking people right out might have results you don't want.
Even if it is allowed the person you talk too might not know it and say no.
Remember bureaucrats will usually protect themselves and give a negative answer even if they don't have a clue about what is true or not, some might actually get off their butt and go find out for you but that is rare.
Most cities have their rules, when I lived in Kansas most all of them allowed it but one city says you can scan for surface finds but no digging.
Can't think of a more frustrating way to do this hobby so I just stayed out of the parks in that city and hit the many others I had all around me.
National parks are usually off limits, some state parks too, but then again being nice goes a long way.
We have a state park here where the rule is only digging in the sand on a small beach is allowed.
However I did go in and talk to the head ranger once and he said as long as I don't wreck up the place he didn't care where I dug.
Use common sense always...it has happened that public parks that had been ok to hunt for decades were all of a sudden declared off limits simply because one new guy went in and asked about it which drew attention to the hobby and someone with power who did not know how small our holes are or how well we cut plugs and leave no trace assumed the worst and just outlawed us.
Rare but it has happened.