The awnser for this question is both yes and no. Most quarterbacks do NOT call plays because they are both impatiant and inexperianced, however some experienced qb's do. A few examples are Payton Manning, Tom Brady, And Brett Favre. Sometimes though they can only call the type of play. For example on first and ten a team (vikings) go for a running play. It is up to Brett to chose to run in the middle, left, right, ect.
They can if they chose to.
I heard that he retired because he wanted to call his own plays as most quarterbacks did back then. Tom Landry was one of the early coaches to call offensive plays.
They Call the plays sometimes and tell the players what to do and how to play
Don Shula and Dan Marino
The most commonly used football hand signals for plays include signals for running plays, passing plays, audibles, and formations. These signals are used by coaches and quarterbacks to communicate plays to the rest of the team without the opposing team knowing the plan.
Setter
Any player on the field of play or the head coach of the team may call a timeout.
For instance, let's use the Vikings; With 3 head coaches. The have 8 offensive coaches, including quarterbacks and wide receivers. 7 defensive coaches including line backers, defensive backs and line. And five other staffs/coaches for conditioning and strengthening.
Motions.
yes
It depends on the team ... Coaches, offensive coordinators, and/or the quarterback call plays. On the defensive side there are formations called, the equivalent of plays for the defense. On the offense, if a play is called but the defensive formation is not conducive for that play, the QB can call an audible at the line.
Coaches are always the boss. They tell you what to do, where to go and how to play. Sometimes, pitchers can call plays as well, but for the most part everyone works as a team.