Yes, the spray is usually attached to the lid. Our local funeral directors all have lid racks that look like big metal mug racks. They are expandable like an accordion and have rubber covered clamps that attach to the lid. They often bungee cord the rack down to the handles of the casket in back, also, to balance the weight of the spray.
Florists have different ways of doing them, but I'll share our way: We use a full sheet of green styrofoam and an 18" racquette. The soaked racquette is placed in the middle of the sheet and three holes are made into the styro on the top and on the bottom of the racquette. We twist three chenille for each of the three holes and pass them over top of the racquette, down through the holes and twist them in back. The funeral director impales the stryofoam onto hooks on the rack and twists the chenille to secure it. I usually cut 3" pieces of hyacinth sticks along the bottom of the racquette to help support the weight (so it's not all just hanging on the chenille).
From that point, I just start greening and flowering right into the raquette. I have a 5' space with a small shelf that I work at. I rest the top of the styro on the shelf, which allows the racquette to drain while I'm working. I made one late this afternoon and stood it almost upright against the wall on a shelf in my walk-in. By tomorrow when it's delivered, it won't be dripping.
I know florists that tube all the flowers, but I think that takes much too long. This works for me. I have to make another one tomorrow, so I'll try to take pictures of the mechanics for you.