Sounds like some fun things to do. Unfortunately, getting from the airport to Hollywood is not easily done via our limited Metro. Literally, to get over there you would need to take a shuttle from LAX to the Green Line station, then the Green line all the way across L.A. to Union Station downtown, and then the Red Line up into Hollywood. If you take the flyaway to Union Station you could shave some time off that. All told you are looking at 1:30 - 2:00 hours of travel time and a bunch of walking once in Hollywood.
Here is a sample routing from MTA. I think they are being generous in saying that it would only take you 1:15 to get to Vermont/Sunset. That of course requires hitting your connection downtown just right.
Not to mention that "jetting" over to Pinks requires a car, the bus, a taxi or teleportation, since the metro doesn't go very near Melrose. The Nearest station is Hollywood and Highland. The walk down to Pink's wouldn't be too bad at about 1.6 miles and all slightly downhill. Of course, that means when you go back to Hollywood and Highland you would be walking all slightly uphill. You could walk it, of course, but your schedule is so tight. Really reconsider renting a car for your limited time.
Another thing to consider is that Hollywood, other than the immediate vicinity of Hollywood and Highland and the areas near the Hollywood sign, is not a great neighborhood. You might want to consider an alternate strategy of Pricelining an airport hotel and taking a cab or bus even to Santa Monica, Venice or Manhattan Beach for the day. This would be much easier to accomplish and probably more enjoyable.