Three-foot running lane question.
 

BR in running down to first base right down the line. One foot in fair territory and the other in foul territory in the three foot running lane.
The throw hits the Batter-runner.
Is she considered running in the lane even though the ball hits her in fair territory?
NF please.

I assume NF means you are looking for Federation rules. Unlike ASA where it depends on where the ball actually hits the runners, I believe NFHS demands the runner have both feet within the lane, including the lines, to avoid an INT call.

Of course, all other aspects of the rule must be fulfilled for the violation to be called.

NFHS 8.2.5
The batter-runner shall be called out when:

She runs outside the three-foot (0.91m) land and, in the judgment of the umpire, interferes with the fielder taking the throw at first base (there must be a throw); however, the batter-runner may run outside the three-foot (0.91m) lane to avoid a fielder attempting to field a batted ball. A runner is considered outside the running lane if either foot is completely outside the lane and in contact with the ground.

Fed: "A runner is considered outside the running lane if either foot is completely outside the lane and in contact with the ground." (NCAA's rule is very similar.)

I assume that should be interpreted as meaning "running with either foot making contact with the ground outside the lane."

BR is running to 1B with his right foot hitting the ground in the lane just inside the foul line, but his left foot is landing well on the fair side. Most of BR's body is thus on the fair side of the foul line. The throw from F2 hits the BR in the back 18 inches on the fair side, but at the moment of contact, the left foot is in the air and the right foot is on the ground in the lane.

I think that should be INT whether or not a foot is touching the ground outside the lane at the moment the ball hits the BR.

Once I was BU in an ASA game in which the BR was running with both feet in the lane but his left arm over fair territory. The throw from F2 hit the BR in the arm, definitely over fair territory, and the defense wanted INT. The PU went by where the BR's feet were and did not call INT. That seemed right to me, but I admit I'm not sure exactly how to interpret ASA's wording "runs outside the three foot lane over fair or foul territory."

In the first case, yes, you are interpreting correctly the NFHS and NCAA rules.

I'm not sure why you have trouble with the ASA wording. The added "or foul territory" only clarifies that the 3' running lane only extends 3' into foul territory. A runner that runs 5' foul and interferes with catching a throw coming from foul territory is as much out of the running lane as one running completely in fair territory that interferes with catching a throw from that angle. The difference in the ASA interpretation is that the interference must occur with the part of the body that is out of the running lane, and doesn't consider it interference if the runner is pegged with a throw in the part of the body which is where it is belongs.