I have done all of the above at one time or another. I've parked at the Boardwalk to eat lunch and am always sent to the first lot on the left after passing through the gate. There has never been a time limit.
I've also parked at the Beach Club to eat there, and again, no one has ever mentioned a time limit. The guard did mention that had they been just a little bit fuller in the parking lot he would not have been able to allow me to park there at all, but he did let us in.
Also, I have walked from the parking at the Boardwalk all the way to the International Gateway and it is a hike. You can take a boat from right in front of the Boardwalk to Epcot, but unless it's already there, it can take awhile. I would suggest parking at Epcot if going to Epcot. Ultimately it is the quicker thing to do because you have the tram.