#78 lines up as a back. Is he eligible to (A) run the ball;
(B) catch a pass behind the LOS; (C) catch a pass beyond the LOS?

Yes in "A" Number 78 can run the ball as a back.
In "B" the ruling would be "Illegal Touching" by an Ineligible Receiver resulting in a 5 yard penalty and loss of down.
In "C" you have "Offensive Pass Interference" resulting in a 15 yard penalty from the previous spot and loss of down.

A) Yes
B & C) Yes, I can think of some situations where this would be legal. But illegal in some circumstances.

The only way he can catch a pass is if B first touches the ball. (unless I'm missing something). If the pass is tipped by an eligible (A) receiver #78 may not touch the pass but he can legally contact a B opponent trying to intercept the ball without being called for DPI.

What other ways are there?

Ineligibles can catch a backward pass any time.

One thing to remember in the B & C situations is that since #78 is an ineligible, he may not go beyone the neutral zone or else we will flag for ineligible downfield.

We had this last week in a game. A team would puit #64 in the backfield for a full house formation. They just used him to block. Everytime he was in the backfield they ran the ball.

You've got it ljudge, tip by B, backwards pass and illegal forward pass can all be caught by #76.

The expanded Neutral Zone...

An ineligible may only go beyond the NZ to the limits of the ENZ if he is in the act of blocking.

I believe the neutral zone only expands for A legally on the line blocking B who is also on the line. With #78 in the back field, he may not go beyond the neutral zone until …

Dumbref, I believe you are correct, as in casebook 2.6.2 it says the NZ is expanded "to allow offensive linemen to..."
Since #78 was not a lineman, he is not allowed to be in the ENZ. Good luck noticing that one though.