I can try to answer some when I worked for a couple of airlines before so it may still be the same.
1. Yes status and fare do matter when in this case. The same as in upgrades generally.
2. No they are not and you have to raise the issue with the cancelled airline about that and they would do their best to wiggle out of it. Again status matters. For eg. you might be booked on premium economy and if the new flight does not have PE, they will put you in normal economy and not business and would not give any extra payments. If you create a fuss probably some points. yes if you have paid for a business fare, they would try to put you on business if available on the new flight.
3. Yes they get paid and normally it is more than what a discounted fare the passengers has paid to the cancelled airline. One reason why airlines try to wiggle out if it too.
4. No, normally they would not but also the passengers does not have to bear the cost of it either.

Thats is what I know and it may have changed now and other airlines might have different scenarios now.