That's what side cutters are called 99% of the time I've heard talk of side cutters.
2nd part, that depends. If your solder removal is adequate, you don't have to destroy the component. Even with a REALLY good solder sucker, sometimes you have to go Joe's route. If I am removing a component I want to keep I'll do the solder suck thing then "wiggle" the component. Not enough to separate a via or trace but enough to see if it's loose, if not I'll heat and suck again. At this point heat damage to surrounding components, traces and vias is possible, so I do it quick and leave the sucker on it longer to drag cooler air over the area. Gary, there really are no hard and fast rules, just observe how it's going and make the next call based on that.