I need to transfer from Terminal 2 (Air Canada) to Terminal 6 or 7 (United flight to/from Hawaii). Do I have to clear security or is there an "inside passage"? Is it close enough to walk?

Also, on our return we have a layover of about 6 hours. Is it practical to go to an airport hotel for a break, or would we be pressed for time clearing security and shuttling, etc?

Sorry there is no inside passage between these two terminals. Just take the "A" bus that travels from terminal to terminal from under the blue sign outside baggage claim.
You could leave the airport for a couple of hours, security is pretty effcient,but I don't know where you would want to go.

Unfortunately, there's not much surrounding LAX. Depending on what time of day or night - you probably don't want to be roaming outside of LAX at all. Just plan on finding a comfortable seat and having something to read. That's about all you have time to do.

The terminals at LAX are arranged like a "U" and in the middle, there is a restaurant called the "Encounter". It's a good place to kill a couple of hours.

If you don't mind spending about $30 in cab fares, you could spend time in Manhattan Beach which is about 10-15 minutes south of LAX. Within three blocks of the pier are several restaurants, an ice cream parlor and shopping.

There are many airport hotels and Marina del rey (great restaurants) is only a few miles down the road. Catch a cab during your layover and get out of that airport!