yes you can keep the old air handler..

if your compressor has hard time starting, yes it mean its aged, a simple but effective way is add a hard start kit to give little extra torque to compressor help it kick in, this is a 15$ fix.

if you decide replace the condenser, you can keep the air handler, a-coil can use across brand, but not across refrigerant, in you decide goto 410a, you need a new one.

.  Since you said you have a air handler but not a furnace, tells me you in southern state where heat is not important.

a 3 ton air handler can't be that big taking 5x5 ft in space, usually its about 30in wide x 36 in deep x 50 in tall. 

air handler is no more than a cabinet housing a blower fan with a-coil inside, you should able find one fit in that space, or go with a regular furnace ..