The door has a seal on it. You'll need to put a stiff putty knife to it and try to lever it off. Over the years, those seals (really, just the sealing wax) gets like a rock. Once off, scrape it clean and re-seal with some silicone caulk.
As for access....well... you could cut a hole behind the "battery tray" in the nose. Just big enough to get your hands through (and will double to pull the shift rod out). Cover back over with something decorative.
As for the coupler, there shouldn't be any play at all at any of the connections. If you have an original rubber one, it might be time to change it anyway. Do not get the urethane couplers - they're junk. There is a guy on Samba who sells billet couplers. I put one on the red car. Gave a wonderful solid shift feel.
If you pull the shift rod, you can then change your shift rod bushing, which may be causing some of your issues. It's fiddly work, but well worth it. The bushings are the first thing to fail in the system.