Hi A340x
If you do a search you will find load of threads on the further education versus flight training debate and you may find these useful. Having a background in Mech Engineering will certainly help with the exams but my personal view is that in the current recruiting climate the carriers you have mentioned wouldn't really mind if you have a degree or not. So with that in mind why not consider saving 5-6 years of your life and go do what you ultimately want to do, i.e pass your flying exams and get a poorly paid but hugely rewarding job flying something, then start moving your way up the food chain towards the jet that you aspire to.
To be responsible, I will add the caveat that obviously this is very limiting educationally and if for any reason you didn't achieve your aim you are stuck with far less marketable education in the broader sense.
Asking people to predict the market in six years time is all but impossible as it is changing all the time and at an unprecedented rate. Also, whilst it's great to have specific aspirations MAT, RYR, EI, be prepared to change your plans to take advantage of whatever opportunities come your way, but I'm certain that you would have worked that out for yourself.
Can't answer your questions regarding Canadians working in Europe but a quick internet search should answer that and you'd be very welcome.
Good luck, keep the faith and you will get what you deserve.