Hola CL -
xxx
Getting a JAR licence and rated on 737 with 1000 hrs on type - no problem.
Whenever they will have openings, they will definitely consider you.
Now, with your papers...
xxx
Your wife has a EU passport. So she has right to live/work there.
On your side you must apply for a work permit.
You will receive (I think immediately) a residence permit (because of your wife being there).
You will receive an ID card (that is the residence permit), usable for travel in EU countries.
After some administrative delays (?) you will receive a work permit.
And as long as you are not a naturalized EU citizen, take your current passport with you.
xxx
As to EU citizenship, in your wife's country, it might take a few years.
Years of being married and resident of that country.
EU citizenship is not required. Residence (ID) and work permits are.
xxx
Applying for a EU airline position, make it clear to them of your status.
(1) EU resident (2) Applying for work permit (3) JAR licence 737/1000 TT...
Based on the above, they are likely to accept to consider you.
Or... they might not. Depends on each individual country laws.
Sure, there are EU multilateral agreements, so do not be too concerned.
xxx
Being native of Belgium, I gave you info on what I know of Belgium/EU laws.
Within EU, legal resident of country X, and working in country Y is quite common there.
Happy contrails