I have a 2002 Chevy Silverado with several issues and I wonder if they are tied together somehow. First the keyless remotes quit working, I replaced the batteries and tried reprogramming them which didn't work. Secondly, after cranking the engine it wouldn't come out of park for a few seconds. The last issue is every once in a while when I try to start the engine nothing happens, it may take up to 10 tries before it turns over. All I get is a faint whining noise, no clicking or nothing. Any suggestions?

You have 3 distinct problems with your car.
1) keyless entry is a " always on " function.

2) Engine not coming out of park is not what's happening. That's transmission related. Sounds like worn linkage. Or fautly cable.

3) Engine not cranking. That could be a starter. The light whirring sound you hear is most likely a fuel pump. The pump normally comes on when the ignition is in the " on " position.

You need to make sure your getting power to the starter soleniod when you turn the key switch to " start " position. If that is happening replace the starter.

( side note ) before doing anything clean the battery terminals really good. And disconnect the battery when replacing or accessing anything electrical on the car.