from what I have seen, taps usually crack on the way out and break later. It is the chip that gets in behind the cutting edge that binds during reversal.
OIL, lots of quality cutting oil. Followed by a good bath of solvent, a fine paint brush to clean off the tap. If I do this every quarter turn I seldom break a tap. I would also seriously invest in a tap guide. Not a block, a guide. They look like a cross between a arbor press and a drill press. I don't have steady enough hands to use a block.
As for getting the tap out. I couldn't add any valuable advice. so, that said, there is always drill out the tap, weld up the hole, polish the weld and re tap. The is a last resort thing though.