All the time. Most are owned by companies with ties to the NFL. If the NFL says "don't show that other league", they won't.
You can be a national minor league with low travel. That’s what divisions are for.
Not as a D league. There's not enough money in it. The NBA has a D league. The only place i see it on television is on the NBA channel. That's why I think the USFL would be smart to partner with the NFL and get a contract to show games on that channel. But it will be stripped down if it goes that route. Fans will know there won't be big stars playing in it.
The NFL isn’t going to say “don’t show other leagues”, that would cut their own throats. Especially since the arena league is already on Fox Sports regional and NBCSports. Throwing their weight around trying to limit what the channels show outside of NFL season would alienate their biggest paying customers.
Yes you can be non-regional as a D league. Remember the NBA D league is actually OWNED by the NBA. It’s very different from the other D/ minor leagues where teams might be owned by major league teams but the leagues themselves are independent to varying degrees. And subsequently can negotiate independent TV contracts.