Player B1 has blood on his elbow, which is noticed by the official. Official sends him to the sideline to get it taken care of. As a result, he is going to have to sit out 1 play. A lines up, and A55 commits a false start penalty which is accepted by B. Does this count as the 1 play that B1 has to sit out for blood?

I don't think a dead ball foul constitutes one play...

I agree with tpaul, but...

Does a blood penalty require one play out like an injured player? I think for an injured player an official's time out is called, but for equipment problems or blood is the player just sent off and thus could return if corrected?

Yes, he is considered an injured player, so he sits for one "down," see rule 3-5-7-c. A down starts with a legal snap or a kick if during a free kick down, so the FS means he stays out until we have a down.

3-5-10-c states that a player with bleeding is considered to be an injured player.

HOWEVER, 3-5-10-a states that a injured player must only be replace for a down if the ready for play is delayed or the clock is stopped for the player. If the clock has not been stopped for the bleeding player, then the only restriction for return would be that the bleeding and/or blood on the uniform has been fixed.

Case plays for 3-5-10 state the clock stopping is the key.

Actually, if you look at case play 3.5.10.B and C it does not matter if the clock is stopped as much as if the "official is involved" in the situation in any way. Good point though if the player notices and not an official at all how it is possible for him not to miss a play.