Is there such a thing as an ineligible runner in football?

Question

In American football, there is such a thing as an ineligible (pass) receiver. That is, a "lineman" (center, guard, tackle) is not allowed to catch forward passes. Are they similarly prohibited from running with the ball? I can see two cases:

1) A "back" (or pass receiver) drops or fumbles the ball, a lineman picks it up and runs with it.

2) The quarterback (or runner) hands the ball to a lineman.

I see the second happening in the case of a "quarterback sneak," that is, the quarterback allows the center to keep the ball and "plunge" for short yardage. Does this mean that a runner can hand the ball to a lineman if it is convenient? Why is this not done occasionally to "mix things up" if it is allowed? Or is such "mixing things up" something rules are designed to prevent?

Answer 1

No, they are not prohibited from doing that.

The rule you stated applies only to forward passes. A Hand-Off is by its definition not a pass, since the ball is not advanced (forward), but handed over instead. That said, any player is eligible to take it.

Going by the following rule they are also eligible to pick up the ball from the ground.

Article 4: Legal Snap.
A snap is a backward pass. The snap must be received by a player who is not on the line at the snap, unless the ball first strikes the ground. If the ball first strikes the ground, or is muffed by an eligible backfield receiver, or quarterback under center, it can be recovered and advanced by any player.

And here is a video of a lineman scoring a TD after taking a Hand-Off.

Answer 2

Fundamentally, a lineman is not allowed to go more than one yard past the line of scrimmage for blocking, or other purposes.

There is a major exception to the rule, which is related to other, minor exceptions.

Major exception: If a lineman begins a block against a defender no more than one yard past the line of scrimmage, he is allowed to continue blocking that (and only that) defender as long as he maintains contact.

Minor exception: From the above, it follows that a lineman, such as a center on a quarterback sneak, who began "operations" no more than one yard past the line of scrimmage is allowed to continue such an operation, such as running. Moreover, most handoffs take place no more than one yard past the line of scrimmage.

Minor exception: If the ball becomes "loose" anywhere on the field, anyone can try to pick it up and run.

Minor exception: No forward passes are allowed past the line of scrimmage but a lineman can catch a lateral (backward or sideward) pass anywhere, and run.