In many (most?) cases you can. But you may need a permit to enter some areas, which may or may not require a fee (if it does require a fee, it's usually pretty small). For instance https://www.nps.gov/yose/planyourvisit/wpres.htm

Usually these fees are for multiple days. For Yosemite: car fee is valid for a week(there are other deals if you want to stay for longer or visit multiple times an year). The wilderness permit... I don't really know. You do have to submit start and end dates for each trip, but the fee covers the duration.

EDIT:

Also Yosemite: it appears that foot traffic is charged too. https://www.nps.gov/yose/planyourvisit/fees.htm

That highly depends on the park.

I cant answer this exactly but suspect “no” since backcountry permits serve to limit human presence more than to generate revenue

Most parks, yes. Some parks may require an additional permit, and I'm not sure if that's waived.

I'm not 100% sure but I think something along these lines applies

I'm pretty sure the answer is: "It depends." Some parks don't even have fees. Plenty of others you can walk into without parking within the park boundaries--this is the case with many areas within Acadia for example. Others you could potentially backpack into but you'd require a backcountry permit which would probably require you paid the entrance fee. Others do have an entrance where walkers still need to pay--at least at the primary entrance. This is the case at Zion as I recall.