Can the kicking team advance the ball on an onside kickoff after it traveled 10 yards?

Answer 1

The free kick is considered one of two types of kickoffs even though it uses a drop kick, according to the NFL rules:"Once the ball is touched by the receiving team or has gone 10 yards, it is a free ball." Conversely, there are the same penalties if it goes out of bounds so it risks a 30 yd penalty.

Yes. The rules for a free kick after a safety are the same as the rules for a free kick after a touchdown or field goal.

Agreed. Just two points of clarification on the original answer:

1. The free kick after a safety isn't a drop kick. A drop kick is a specific type of kick in which the ball is struck after it bounces off the ground. What you usually see after a safety is just a punt without a line of scrimmage in front of the kicker.

2. An onside kick that goes out of bounds doesn't incur the 30-yard penalty. The receiving team gets the ball at the spot where the ball went out of bounds.

Answer 2

Yes, but most the majority of the time they knee the ball.