1: Per the NFL Rule Book, Rule 8, Section 1, Article 1:
Definition. It is a forward pass if:
(a) the ball initially moves forward (to a point nearer the opponent’s goal line) after leaving the passer’s hand(s); or
(b) the ball first strikes the ground, a player, an official, or anything else at a point that is nearer the opponent’s goal line
than the point at which the ball leaves the passer’s hand(s).
Then in Article 2 (emphasis mine):
The offensive team may make one forward pass from behind the line during each down.
It goes on to note in Item 1 that it is illegal if it is
[a] second forward pass thrown from behind the line of scrimmage.
So for your first question, the ball must be moving closer to the opponent's goal line than where it started in order to be a forward pass. Only one may be attempted each play.
2: Laterals are considered to be backward passes. Rule 8, Section 7, Article 1:
Backward Pass. A runner may throw a backward pass at any time (3-22-4). Players of either team may advance after catching a backward pass, or recovering a backward pass after it touches the ground.
Reverses are not related specifically to forward or backward passes. Rather, they are "common trick play[s]... that involves one or more abrupt changes in the lateral flow of a rushing play" (Source). More often than not these will start with a hand off, not a pass.
So backward passes are by definition a pass between players that does not advance forward; reverses are a type of trick play.
3: Every reference in the rule book about the who an eligible passer is says either "the passer" or "the runner." Specifically, Rule 3, Section 27, Article 2, Supplemental Note 2 states:
The statement, a player may advance, means that he may become a runner, make a legal kick (9-1-1), make a backward pass (8-7-1), or during a play from scrimmage, an offensive player may throw a forward pass (8-1-1) from behind his scrimmage line, provided it is the first such pass during the down and the ball had not been beyond the line of scrimmage previously.
In Article 1:
The Runner is the offensive player who is in possession of a live ball (3-2-1), i.e., holding the ball or carrying it in any direction.
Any player who is deemed a runner is eligible to pass. This includes most offensive players except for the offensive line (i.e.: center, guards and tackles). Players would have to report in as eligible receivers in order to be considered eligible to pass. For example, if an offensive tackle report in as eligible, he could technically pass the ball (although this seems like an awful idea).
Also note that if a ball is fumbled behind the line of scrimmage, an offensive lineman could pick up the ball and commit a forward pass. But again, this seems like a terrible idea.
Response to comment:
This is where it technically gets a little fuzzy, since the phrase "hand off" rarely appears directly in the rules outside of citing examples. It does say in Rule 3, Section 27, Article 2
A Running Play is a play during which there is a runner and which is not followed by a kick or forward pass from behind the scrimmage line. There may be more than one such play during the same down.
So a hand off counts as a running play, but so does a backward pass. However, the rules explicitly state that a hand off cannot be forward if the ball is beyond the line of scrimmage (8.7.4.a):
No player may hand the ball forward except to an eligible receiver who is behind the line of scrimmage.
So once the ball passes the line of scrimmage, the ball can only be handed or passed backward, never forward. But behind the line of scrimmage, the hand off can be forward or back and it is not considered a forward pass. This is why a muffed hand off results in a fumble rather than an incomplete pass. Additionally, a muffed backward pass is considered a fumble as per Rule 8, Section 7.
