For example, Team A is down by a score late in the game, say third-and-ten at the opponent's 30 yard line. A pass play is called and the quarterback throws the ball to a receiver with a gain of say 13 yards, and the receiver thinks it's a good idea to lateral the ball to a teammate behind to gain additional yards before he is tackled, however the recipient is then tackled 5 yards behind by the defense before he could advance, the next down will start at where the ball is blown dead so it's 4th and two for the offense? The gain of first down from the successful pass will be negated since the ball is thrown backward right?
You are correct; no first down would be awarded, and the down would be 4th and 2.
You can see this from the NFL Rulebook, Rule 7, Section 3:
ARTICLE 1. NEW SERIES FOR TEAM A
A new series of four scrimmage downs is awarded to Team A when the following conditions exist:
During a given series, the ball is declared dead in possession of Team A while at or beyond the line to gain.
There are several other reasons for a new series of downs (meaning, first down) is granted, but they relate to things like penalties or changes of possession.
The only check for first down is at the dead ball spot - it doesn't matter what happens during the play, just as when you go back behind the goal line. (Scoring a touchdown, however, does occur instantly upon crossing the goal line.)