Is cleaning the tub with bleach bad for the plumbing?
Bleach is really tough on surfaces and over time may damage the tub. If its diluted, I don't think that it will affect pipes, but then again, just about anything is bad for 100 year old iron pipe.

After you get the tub clean enough, I'd recommend using one of those daily spray tub/shower cleaners or a steam cleaner like a Haan. The bad thing is that you need to use these products more often, but the good thing is that it takes minutes and little work vs. scrubbing and burning your lungs.