Can people legally live in a national/state park or forest?

Nope.

No, you can not "live" in National and State parks. They are public lands for the use of everyone. Some Parks have designated caping areas and some even have what they call "Full Hook-ups" which means they have Electrical, Water and Sewer/Septic service for those sites, but there is usually a Maximum Time you can stay in one park for consecutive use. The rates vary depending on the improvements and the particular park and sometimes vary with the seasons, but most State and National campsites I have used run about 10-15 dollars a day. Some low usage parks will allow Quasi-permenant sites for people that volunteer to act as Park Hosts, but this is usually a seasonal thing.

it is possible for people to be long-term campground hosts, but they technically don't live there in the sense that they don't have a house or permenant address, usually its a retired couple that lives in an RV

You can also camp or go backpacking etc for long periods of time,

However no you cannot just build a house and set up a permenant residence.

Plus you need to distingush between the three things you have listed:

National Parks are highly protected areas

National Forests are nationally owned but not as tightly protected (some private companies are allowed to log certain areas by permit, ski areas are allowed to opperate with permits etc)

State parks are owned and opperated by each state, each state sets its own rules and they are entirely separate from national parks and national forests.

The answer is a certified maybe, I grew up 5 miles from Sam Houston National Forest and a lot of people had homes there. On the other hand you cannot set up housekeeping in one of the campground facilities or the Rangers will evict you.

With specific guidelines, yes.

Where I live there are homes/private residences (not including park employees/students doing work/research in or for the park)

inside the national and state park boundaries but no new structures are allowed unless authurized by the park systems, and I believe they have to beneficial to the park. We have an old structure being updated to include a farm for local produce stands. The approval on this took years.

It depends upon what park/forest you're talking about. There are homes in some national parks/forests that were there when the land was designated as such. The families who owned the land generally can remain for a given period of time. They no longer own the land itself.

Absolutely, and as mentioned before, park rangers are a prime example of this. I also lived in one myself for about 6 months, in which I worked for the park and was given residence and accomidations. Its a great experience actually.

There is a Park near where I live that homeless ppl have lived for years and today they tore down their tents and made them leave.

Yeah. Park rangers live there all the time.

Nope, that is why you are only allowed a certain amount of time in one campsite before you have to move.