Depending on the style of the cage, he may be able to safely go a week without removing it to do a thorough cleaning, but should never go longer than that.
If the cage is not one that is a wire style which allows sufficient water flow for cleansing, then he needs to remove it for cleaning at least every other day, unless you wish to risk him developing a rather nasty rash. Been there, done that and didn't even get the lousy T-shirt.
You can always handcuff or restrain him in some fashion to insure he won't masturbate while you clean him or after he cleans himself under supervision if you are permitting him time to "air dry", which you should do to allow his skin to breath, especially if the capture ring that goes around his testicles is the wide hinged type.
I would definitely recommend that you do not use his sanitary needs as a form of punishment.
As far as how often he can go without an orgasm, the school is out.
There are men in various cultures who do not permit themselves to orgasm and have had no ill effects. Other schools of thought say it is necessary to keep all the "plumbing" in healthy working order. As you say you are "milking" him, that should serve to take care of the "plumbing" issue without allowing him the pleasure of an orgasm.
Many chastity practitioners actually find that having an orgasm after a long period of being locked up to be somewhat anticlimactic as they reach a plateau of sexual frustration and desire that is itself a desired state of being and complain that it takes too long to regain it after having an orgasm that lasts only a few moments.
This even though they can't help themselves from wanting the orgasm that takes them out of that heightened state of desire.
As your bull now controls when he is released, he apparently won't need to worry about this either, unless you choose to include it as a punishment. Imagine that, being ****** to orgasm as a punishment!
Actually, ****** orgasms can easily be a form of punishment. Although much easier to do with a female, forsing a male to masturbate repeatedly until he can no longer become erect and achieve orgasm can almost become a truly painful experience for him.