Bottom bracket axles come in a variety of widths as do bottom bracket shells. Typically these measurements are marked on the axle or cartridge and you need these measurements to get a replacement. Needless to say you need to take a fair amount of parts off the bike to get to read the axle/cartridge, however that is the best way to get the required numbers.
In the olden days when steel was king it normally came down to where the bike was from to get the bottom bracket width but a lot has changed since then and there are no convenient rules of thumb. That said, with a micrometer or even a ruler you can measure the bottom bracket shell width.
In these simpler times a given bottom bracket shell size would come in up to three axle lengths for 1/2/3 chainrings - the longer axle needed for the 3 chainrings.
You can work out what you need for a given bike thanks to the internet - Shimano provide their specifications online as do the less common brands.
As for chainring spacing that does not really of concern here - your LBS got it slightly wrong but you can forgive them because you have to 'get a few wrong' before replacement bottom bracket axle/shell lengths are understood.