I've done the job a couple times. You need this stuff:
Mobile Home/RV Black Butyl Tape 3/4" x 30' for Windows, Doors, Roof Sealant | eBay
I'm sure the tubes as Pntyrmvr suggested are the same but I thought this was easier to work with. It's what SMB uses.
Pull your upholstery back from the inside of the top with top raised and you will see 1" furring strips around the inside screwed every inch or so. Pull all screws and strips which are probably rotten to pieces. You'll see the flat butyl tape underneath. You just need to remove everything you can, lay in new tape, furring strips (any 1/4" ply will work) and reinstall. The furring strips will squash the canvas into the butyl and form a seal. I have not heard of one leaking like yours does. Even with rotten wood they usually stay mostly dry. You could do this job in a day, no problem. There is no need to mess with the lifting mechanism of the top at all. Raise it, replace everything I listed, reinstall. You may need to pull the top down just a bit to give your canvas some relief when reinstalling strips to get a good compressed seal.
This is a messy job. Try to keep the butyl gunk (mostly from the old tape) off your canvas and upholstery. It likes to stay on what it gets on.
As for the drip at the rear center of raingutter Twogone had the same problem and solved it ingeniously. Get a roll of gaffer's tape (or any other comparable tape but it has the right texture/thickness/etc.) and run a strip across the back of the rain gutter above the doors. Fold it perfectly where it creates a sweep that the doors go right under when closed, but the tape stays in place, catching water and it then drips off the edge of the tape. I'm probably not doing a good job of explaining this. Maybe he can help.