Unless the weights fell off, only when you disturb the tire from the rim.
If they are already balanced then no you dont. but if they are not balanced yes you do. If you dont you will wear out the tires faster than if they were balanced.
when they are no longer balanced.
Yes, you should have balanced all 4 tires.
Balance the tires. That will most often solve the problem. There COULD be other issues, but don't worry about them until you've made certain that the tires are balanced.
Tires are out of balance. Have the tires balanced.
Yes
very
If the tires are properly balanced plug tires should not cause a vibration.
All tires should be balance checked before being placed on a vehicle. If the tires are not balanced it will cause uneven wear and they will not last as long as they should.
There is no simple answer. Tire life is affected by many factors. To make a tire last make sure your tires are balanced, rotated often, properly inflated, and the alignment of the vehicle is good,
Could be several reasons but the first one that comes to my mind is to have the tires balanced.
Check your tires. I had the similar feeling after I had my tires rotated. Come to find out the tires were not balanced.