So I'm putting together this old Royce Union as a beater commuter. The bottom bracket has some play in it (not sure if it's just loose, or trashed). Also, the cranks are cottered.

I have NO idea about anything bottom bracket or cranks. Are they interchangeable (cottered and not)? I had considered swapping the cranks so I could use a different chainring... I have a small collection from when I raced BMX bikes. Wondering if i should/need to swap BB and cranks together. Any idea on a REALLY cheap set? It's a single speed now, and will have a flip flop hub.
What measurements do I need? I mean, I know the width of the BB shell. What else? Distance from center to the rear cog? BB diameter, where would I measure that?

Install myself, or just take it to a shop to have it done?