The data should always be king--not the math--inconsistencies should point the finger at the math and theories, not the other way around.
Except when the data are "bad". The most recent example of this was the Opera experiment's claim to have observed faster that light neutrinos. That's what their data said...at least until someone found that their GPS cable was loose.
Experimental science is never that cut and dried. The data may always be right given the experiment performed but, as the Opera case shows, that may not be the experiment which you think you had performed. The result is that there is always a tension when theory and data contradict: is it because the theory is wrong or is it because of an invalid assumption when interpreting the data?
No, new theories do not have to look mathematically connected to the math of old theories--this is the root problem--assuming that the old theories are *proven* and *correct*.
Here you have taken a position which contradicts your earlier one. We believe old theories *because* they are consistent with the data we have so far. Hence, by logical extension, any new theory must also be consistent with that same data otherwise we would point to that data and say "see it disagrees with data and so it must be wrong!". Within the precision of existing data any new theory *must* have identical predictions, i.e. an identical mathematical form, to the previous theory under the conditions where the old theory has already been tested and confirmed. It can only vary under situations where the old theory has not yet been tested.