If the wheels are shaking, then they need balancing, regardless of how many weights may or may not be attached to the rims (you generally have no idea how they behaved on the previous car).
If the car handles funny, steers vaguely, pulls to one side, then an alignment is called for.
Simple.
Personally, I steer clear of fitting S/H tyres. They are generally set into rotating in a particular direction, and if you change that, then they start misbehaving and giving strange performance and handling.