Which will burn more calories -stationary bike

They should be identical. Doesn't matter how fast you do the distance, you'll burn the same number of calories.

Option two, same mileage but more resistance. More calories burned.

*edit*
If you would have done :
10 miles in 30 mins no resistance, that would have been the same as 10 miles in 60mins no resistance.
If you add something to either one, that would burn more.
Going deeper is just a waste of time.

They should be identical. Doesn't matter how fast you do the distance, you'll burn the same number of calories.

Why do heart rate monitors matter, then? It's hard to believe that walking 1 mph for 4 hours burns the same as 4 mph for one hour.

As far as walking vs running there is a difference because running takes extra energy to propel your body off of the ground. You don't have to do that when you walk. As far as different speeds of running, if you run at 6mph for one hour (6 miles) or you run at 8mph for 45 minutes (6 miles), you will burn the same calories. Same goes for riding a bike. It's thermodynamics. It takes a certain amount of energy to do a certain amount of work (moving over a distance). Power (work per unit time) does not matter when it comes to energy.

Wait, I'm confused. Maybe I should say 20 miles in 60 minutes, no resistance
Vs
10 miles, 60 minutes, level 7 resistance

my guess is about the same .

work = Force*distance so although you might be requiring more force with level 7, you are going half the distance.