Is it necessary to divide dahlia bulbs? If so, why?

Answer 1

Dahlias produces many side tubers that eventually crowd each other diminishing the show by growing weak stems and smaller flowers. So yes it should be done every few years to keep the plants performing their best.
Wait til the first frost kills the foliage then wait one week before digging. The tubers produce eyes during this time. If you do not freeze to kill the plants dig them before mid November anyway. Cut the plant down then dig in one week.

It is OK to cut a tuber's end off. They can even be stored that way. Just do not break or cut off the long thin necks. Set the clump on a BBQ grill or a soil sieve and hose off the soil. Working in the shade so the tubers don't dry to much. Trim the roots all off.
Look the clump over to locate the eyes. This is the future growth site so each piece needs one to produce a plant. Eyes are only located near the crown of the clump. Remove the stalk and separate the pieces with eyes.