Hi Mark Interesting challange. You don't mention why you want to replace the existing floor joists...too much sag? You also don't mention what the flooring/subflooring is, but I'll assume its 1 by X pine running at right angles to the joists.
Trying to remove the existing joists while keeping the floor above in tact will be a huge amount of work. You will have to cut all nails holding it to the floor above, and then remove the nail stubs that are remaining. Each joist will have to be done one at a time. No, don't notch 2X12's, as you will end up with the strength equivalent of a 2X6. And if my math is right, 4 2X6's together will make a 6X6, not 4X6. And if you do these on 12" centers, you will end up with a joist system that is 50% 2X6.......somewhat of an ove-kill :-)
I've only done this a couple of times, but have had good success. I put a house screw-type jack in the middle of the joist (putting one on each joist running parrallel on each side of the joist you are lifting would be better yet), and lift the old joist until the sag is less than 2" per 16 spanned feet (less than 2%). If your sag is worse than this, it may take days or weeks of constantly increasing lift pressure to lift the sagging member. Rule of thumb is don't try to raise it more than 1/2" per lift (at least this is what I've been told). Then sister a 2X6 on each side of the old member, holding it in place with #16 resin coated spikes ( a nail gun makes this much easier) and a good construction adhesive (like PL 200).