Old paint bleeds through? ? ?

Shellac based primer. OTOH, if it has that many coats, it probably looks like hell and should be stripped

Did that include Zinser(sp?) sealer?

I've used that titanium white to cover over water damaged redwood, when nothing else would stop the bleedthru.

Have fun stripping the doors down to the wood and either repainting them finishing them some other way By the way, If the doors are as old as the house, you should seriously consider stripping them them.

Bleeding through multiple coats suggests that the source of stain is the wood; most common with "oily" woods, like cedar, redwood. Stain blocking primers should solve the problem...there are plenty of good brands, with Zinsser and Kilz being pretty popular. If the surface is sound, clean it, prime, paint.

Pale pink is suggestive of mildew...does bleach touch it?

I agree. One product is this one made by Zinnser:

Or, I have heard some people say to just use shellac.

True, especially if you are just talking about one or two doors. And then use a shellac-based primer.