I wasn't involved in this show when my group did it, but I've had to do some similar effects.
For the hatch door, try mounting it in sliding rails like you would see on cabinet drawers. I used that for an effect on a show just recently. The rails are cheap, reliable and easy to install. We had our sliding rails for a hidden panel, and used a basic pulley system and a stage hand to raise and lower it. For the one quick close we had, the stage hand would just let go of the rope and let it slam down.
For No Sex, you want to make sure your hatch stays securely open since Runnicles has all that business with the hatch. I once saw a production where they had a badly installed mechanical hatch and it closed on the actor's head during a performance.
As for the flowers... I'm stumped. I don't remember this part of the show at all. I'll ask around and see if anyone from our production remembers.
My initial suggestion is that you could do something with a small bundle of nice, big green straws (like the ones at starbucks)... Take a bundle of straws, tie them together with some fancy fabric and ribbon, and disguise the bundle with a few pieces of artificial greenery... then tuck the flowers and greenery that will be 'guillotined' into each straw... when the hatch comes down, all the heads of the flowers will be on the wrong side of the hatch and you should be able to slide the bundle right off... if you glue a few 'broken' looking pieces inside the bouquet, it should look right... (Does this description make sense to anyone but me?)