It's all about money: teams make money from fans buying tickets, from fans paying for parking, from fans buying beer, from fans buying hotdogs, from fans buying nachos, from fans doing everything else fans do on game days at the stadium. 17 games gives them 6% more game day income than 16 games. 18 games gives them 12% more game day income.
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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 ...