Whoa.....for a minute I thought Pmenu was bought out by T-nation.....Ha!
Whitney I would first have to say that you will probably not find too many (if any) people here doing the old bodybuilding splits routines (aka chest/back, bi/tri). What you will see is more of doing 5x5 routines, Powerlifting, some crossfit metcons, etc. So it's hard to answer your question when you basic workout outline is different from others.
I don't know your goals or current state of fitness, but I would suggest at least dumping the old body split workouts and move to more whole body movements of deadlift/squat/pullups/dips/etc. You will get plenty strong and build size going heavy on the big lifts.
As for when to use pushups....well if they are not part of your routine, I would say put them in when you want and see how your big lifts (in this case it would be bench and overhead press) respons. Have fun with it....do them around the house...at work....GTG type of stuff....since you really can't go heavy on pushups it's more the the GTG or metcon related activity. If your big lifts ever suffer then you may be doing too much GTG stuff. That's the best way to tell if you are overtraining is by the numbers on your big lifts.
Dump the Bi/Tri days...you will get plenty big arms doing pullups, weighted pullups, dips, weighted dips, etc....if you want to throw some curls in here and there, go ahead...but don't make a seperate day for it and don't do so much that your big lifts numbers suffer or make no progress.