As the subject line suggests, I plan on leaving my boat on my lift with a fixed Boat House roof over it for a few weeks at a time. Any suggestions like leave the drain plug in and cover the cockpit with the canvas cover Etc...
I also plan on tying up a couple of lines to keep it from swaying in case the winds pick up. My Lift is a double motor overhead with aluminum bunks. This is on Perdido Key which is adjacent to the Gulf of Mexico

Make sure the drives are out of the water, tilt as necessary.

Plug out or good auto bilge pump, cover on. Tie off, all sounds good. Do it all the time.

I've never stored my boat on a lift for more than a week or two, but if it were me...
...I'd look into a self bailing drain plug for when I leave the boat unattended.

With having a fixed roof, I'd leave the plugs in and cover on. At least that's what I and everybody else do at LOTO.

winner...............and tie it off

Yes plugs in and cover on with support poles in place. If the lift goes down you don't want it sinking. And tie it off to dock with appropriate slack in ropes so it doesn't float away if the lift goes down.

If no expected Hurricanes, I'd be more worried about bird schit.