Crazy component mix
 

These are the components on a bike I am considering buying. Do you think they will work together? Hoew much would I have to relace to get them to work? I am thinking that the rear derailleur would definitely have to go.

SHIMANO Ultegra 9 Speed Cassette

SHIMANO Ultegra 9 Speed Chain

SHIMANO Ultegra 9 Speed Left Shifter

SHIMANO 105 Right 9 Speed Shifters

SHIMANO Sora 8 Speed Rear Derailleur

SHIMANO Sora 8 Speed Brakes

SHIMANO 105 10 Speed Front Derailleur

SHIMANO Compact 50 34 T Crankset missing the crank bolt

Shimano 105 10 speed bottom bracket

Should work fine. With very few exceptions, all Shimano road parts work together so long as the rear shifter, chain and cassette are the same number of speeds.

Sometimes cranksets can be a nuisance if you have a chain thats designed for a higher number of speeds than the crank, because the chainrings can be spaced too wide and allow the chain to slip between the chainrings. I have this problem on a road bike that has OLD (1984) 6-speed campy cranks and shimano 9-speed everything else. An 8-speed crank with a 9-speed chain shouldn't be a problem. In one of his recent articles, Lennard Zinn wrote that 10 speed cranks should work with 9-speed chains as well, so I think you're covered.

Thanks. I am a relative newb to wrenching but I would like to learn more. Can you explain what is meant by eight speed brakes and a 10 speed bb? Does this mean that these components were orginally meant to be part of a eight speed and 10 speed group? I don't understand because they don't seem to be part of the gearing.

Love those 8 speed brakes.
As you progressively pull the lever, the indicator shows:
Hoy Fast
Really Fast
Fast
Moving Along
Casual
F---ing Around
Dawdling
Almost Trackstand
Stop

Just joking. Just about all road brakes work with all road levers.
You need to match the shifters and the cassette, but all your other drivetrain components should be interchangeable, if they're current and all from the same manufacturer.
10, 9, 8 speed just refers to the group they typically go with.