How close is the beach

As previously stated the bay is right behind but if you want a beach with crashing waves and a boardwalk with little shops it's about a 3 mile walk, bike, or drive

Waterfront rooms are on the beach/boardwalk. (F.Y.I... Its the bay..kinda like a channel. But still nice, quiet, and at 9pm every nite you basically have a front row seat to the fireworks that Sea World does.)

For the same money, you can stay in La Jolla which is much nicer and closer to the beach.

Yes

10 feet from the back gate

Very close to mission bay. So, yes a beach, but not the ocean.