Problem #3 is about noncancellation phenomena. Peter Hilton (often with G. Mislin and/or J. Roitberg) wrote about ten papers on the subject. There are plenty of examples, generally related to localization.

Probably a good start:

Hilton, P. Non-cancellation phenomena in topology. Topics in topology (Proc. Colloq., Keszthely, 1972), pp. 405–416. Colloq. Math. Soc. Janos Bolyai, Vol. 8, North-Holland, Amsterdam, 1974.