As jonthefish said.
And, pictures/year would help.
Basically:
bottom bracket - has different length, type and diameter. What you probably need is a square taper (type), diameter should be fine (other type is rare) and length according to the crankset you are planning to use.
crankset - determines bb length, for 3x7 and 3x8 systems, they should be interchangeable.
front derailleur - what you need is correct cable pull (top/bottom) and diameter of the tube (actually seems you have both correct) to be able to mount it and connect with cables. Then if it doesn't reach, might be that the bb is incorrect (way too long) or the derailleur has the limit screw tightened so that it won't reach. Or, that it is some derailleur that won't reach anyway which would mean it is a road type (it probably isn't).
cassette - number of cogs should be same as shifters and rear derailleur need (and chain and rear hub should be also one made for the same), apart from that, most common you buy should be fine.
chain can be any, made for the number of cogs on the cassette. if it skips off, it should probably be about adjusting, as mentioned above.
Good luck!