Yes, there is usually a sticker on the shell that gives the size, but it will often be faded, worn or scratched and difficult to read.

To test the crank, undo the retaining bolts with the allen key by turning counter clockwise on both sides, and see if you can pull the arms off by hand. If you could they were loose, and the crank interface may very well be damaged and come loose by itself again even when tightened. Spin the bare spindle and see if it is loose or rough. If it is okay, clean the spindle and the square hole on the cranks, apply some grease, and attach the arms back on the spindle and tighten down the bolts very firmly.

For the bottom bracket, the driveside is reverse threaded, keep that in mind when trying to undo it.