American football games with just a ball?

Not a problem. Google "Flag Football". It even has no contact rules for old farts like me. You get to run and throw without the beat-down. We always used to play two-hand tag football when I was a kid. everything the same as football except you tag the runner/receiver with both hands instead of smashing him to the ground. We played it as a street game quite often.

Edit: Just saw you don't want to play touch/flag type. Seems to me you need to create your own level of rules directed to how hard YOU want the game to be played.

I played flag football while at Cambridge University. You may be able to find pick-up games at some of the colleges. Most of the other players also played rugby, but as a Canadian, I did not have that background. The ball is ruled dead where the ball carrier's flag is removed. In theory it is possible to get the flag without tackling, but in practice the ball carrier was often tackled in the process of removing the flag. All of the positions including offensive and defensive linemen were included in the Cambridge U version (I played linebacker). I found the British version of the game quite civilized. There was pushing and shoving to get position, but no bone crushing hits.

You can play flag football (American football but two flags - one on each side of the waist - are attached to a belt that every player wears, and a tackle is made when a defender rips off a flag). If you choose, you can let this get a bit rough where people can still push or block hard, even occasionally getting tangled up and tackling, or you can enforce no contact.

Soccer Baseball American football In those orders

When you play a pickup game...you usually have 1 person play as the Quarterback and everyone else is a Receiver. The team on defense normally will have everyone cover a receiver and one person that can stay on the line of scrimmage and Blitz the QB after a certain amount of time. Usually 7 seconds.

It depends on how fit you are.

In the NFL they wear those helmets and gear because players have been paralyzed, and almost killed, even with the gear.

If you and your friends play Rugby then you have nothing to worry about because you won't get hit as hard as the players from the NFL. So just hit away.