bottom bracket interchange?
 

i am new at biking so i am sorry if this question is dumb. i did a search and didn't find much on this subject.
i an restoring a 92 trek 2300 bike. after a long wait my stripe guy finally got me my treck decals. finished painting so on to the assembly stage.
when i got this bike it has been rode hard and put away wet several times. the 8 speed shifters were gone so after comparing prices i found out 9 speed parts are actually cheaper than 8 speed parts.
the shimano crank gears were rather worn so i figured replacement was necessary. so i found a bontrager race light crank with a bottom bracket reasonably priced so i bought it. after opening it up i see the bottom bracket has external bearings and the shaft, with a octalink, looks wider than the original square shaft shimano 600 one that was on the bike.
WILL THE DIFFERENT BOTTOM BRACKETS INTERCHANGE and everything line up?
i know it will bolt on because i tried it. but will it line up?
i have new shimano ultegra 9 speed derailleurs and cassette but will it line up with the bontrager crank?
forgot to add. the 600 crank is a double the bontrager is a triple. if they interchange can i remove the small ring and use the 600 double derailleur or do i need a new front derailleur also?

thanks for the answers in advance.

Provided that the cups are threaded the same (sounds as though they are), the external bearing crank will install properly in place on the old crank. External bearing cranks have a fixed chainline determined by the manufacturer so what you got is what you are stuck with but it should be correct assuming road crank and road frame.

If you had the frame repainted, you will want to have the bottom bracket shell faced to remove any excess paint and square up the faces. Not 100% necessary but it may save you some installation headaches and grief down the road as non-square faces will cause difficult assembly and creaks and quickly wear out bearings.