Should falsifiability be a part of any religious education ?
That would be nice, but I doubt that would ever happen in a religious context. At least in the Christian tradition, it takes 'faith' in order to believe in it. In a religious context, faith is trust in a belief, which takes some justification to believe in it. If you have to justify your beliefs, there will always be doubt.
Falsifiability will draw a line between what is scientifically supported and that which is not. The scientific method, I imagine, would always disprove beliefs that have to be justified (as in "it is so because the bible says so" or "because my priest says so" ).
Besides what would be the name of such a religion? Science- tology? LOL! That one is already taken and what a mess that is!
The only religion that has stated that it would have to change if science disproved something is Buddhism. At least that's what the Dalai Lama has been quoted as saying as head of the Gelug school of Buddhism. I don't think their belief in reincarnation is in danger of being disproven by science any time soon.