but my question is, will a degree in engineering open up any specific doors as a pilot
In aviation “doors are opened” when you are in the right place, at the right time and with the necessary qualifications.
When the economy is booming, like during the .com expansion of the ‘90s, opportunity was everywhere and the major manufacturers were actively looking for personnel it is much easier to achieve your professional goals.
On the other hand during a major economic downturn it becomes difficult to reach lofty career aspirations when bankruptcies and redundancies abound.
possibilities of becoming a test pilot, working for Airbus, ESA, EADS etc.
In general I would say yes; an engineering degree would put you in a better place than a liberal arts degree for you to make the eventual jump from line pilot to test, management, technical or manufacturer's pilot.
Receiving your engineering degree, by itself, guarantees nothing for your future aviation objectives. It helps to be prepared, so on that point I would say that an engineering degree is more valuable.
Attitude, qualifications, experience, motivation, work ethic and education are all important. You cannot control the business cycle or geopolitical upheavals that influence the world economic climate, so it is difficult to place yourself in “the right place at the right time” but you can make yourself the best possible candidate when and if the opportunity presents itself.