Is it legal for a player to receive a pass from a teammate that is in the backcourt, while jumping in the air and have one foot land in the backcourt and one foot land in the front court? This is the ball's first trip to the front court.

Depends when did the player catch the ball?

The player caught the ball while in the air and came down straddling the midcourt line. Again the ball came from the players back court and this is the first trip for the ball into the front court. The question is regarding straddling the line while coming down with the ball.

Where did they jump from?

Which foot landed first?

If the "receiver" had been in the frontcourt when he left the ground, this is a violation. If he had been in the backcourt, it's not. If he had been straddling before he left the floor, he was in backcourt, and it's not a violation.

four requirements for a backcourt violation\

1. team control
2. frontcourt status
3. Team A last to touch the ball before it had bc status
4. Team A first to touch after it had bc status

In your example, the only questionable item is #2. If the receiver jumped from the front court, he has front court status and imparts that to the ball when he touches it while he's in the air. THen when he lands straddling the line, he was last to touch and first to touch, so it's a violation.

If that's as clear as mud, someone will chime in here after while and bail me out.

It is illegal if the player "alights" from the back court catching the ball while in the air, before landing in the front court. The player status is where he left from until he lands where he is going.

I'm not quite sure what you're saying here, but something is wrong.

1) "Alight" means to land. I'm not sure how a player "lands from the backcourt."

2) If the player leaves the floor from the BC and catches the ball thrown from the BC, then lands in the FC, it's a legal play.

To Ege31 -- In the original play, note that the exception in the BC rule (an airborne player may catch the ball and land in the BC) applies only to a THROW IN, not to a pass from the BC. As described, the play is a violation.

I meant if he takes off from back court, catches the ball in the air and lands in the front court, its back court. I am assuming the pass originated from a player in the front court. Maybe alight was the wrong word.

In this case, his landing is irrelevant. He may as well land on Mars, because this is a violation as soon as he touches it.

Thats true, soon has he touches the ball, tweet, backcourt.

The pass, or deflection, or very long rebound originated in the backcourt. The player lept from the backcourt, snagged the ball and came down with one foot in the front court and one foot in the back court. He was called for a back court violation which was incorrect, right?

Hope this helps.
SECTION 9 BACKCOURT
A player shall not…
ART. 1 . . . Be the first to touch a ball after it has been in team control in the frontcourt, if he/she or a teammate last touched or was touched by the ball in the frontcourt before it went to the backcourt.
ART. 2 . . . While in team control in its backcourt, cause the ball to go from backcourt to frontcourt and return to backcourt, without the ball touching a player in the frontcourt, and be the first to touch it in the backcourt.

Furthermore:
ART. 3 . . . A player from the team not in control (defensive player or during a jump ball or throw-in) may legally jump from his/her frontcourt, secure control of the ball with both feet off the floor and return to the floor with one or both feet in the backcourt. The player may make a normal landing and it makes no difference whether the first foot down is in the frontcourt or backcourt.
PENALTY: (Section 9) The ball is dead when the violation occurs and is awarded to the opponents for a throw-in from the designated out-of-bounds spot nearest the violation.

It depends.
If it was a pass from a teammate, then we have team control. If his front court foot landed first (even barely), then we have FC status. Violation as soon as the other foot touches down in the BC.
If it's a rebound, no BC because there's no team control until he catches it. If it's a deflection, it depends on who had team control during the loose ball.