Do I have a chance to transition from a support role to engineering if I stay in the company? Meaning, how often a company will give the opportunity to "the support guy"?
I've known several people who have made a transition from support to a development role. Basically, you want to start by letting your manager know that you have a degree in computer ...
You asked several related questions. Unfortunately, they're all fairly hard for us to answer directly, because they are fairly personal choices. Ultimately, you need to decide what is important to you in a job - both in the short term sense:
Are you happy with the work in front of you? Is it fulfilling?
Do you like your coworkers, your boss, your company?
This answer is assuming you will definitely be moving to City 2 regardless of what happens with your employer, the client, etc.
If it would be a practical and appropriate option for you to work for the client at their 'City 2' location (e.g. their office in City 2 would be a suitable place to carry out whatever your job function is)...
Given the ...
You could become a product owner in a company that delivers software products and head a team of developers with personal focus on organizational work (project management), not development (coding) itself.
Just be honest with yourself and company and this shall come to good. In fact, you shall discuss with your company the reasons behind your decision to move to another city and they would try some middle way if possible.
After taking training or buyout of the contract, it will be a difficult situation for both parties to break up.
what is the ideal career for someone with a master's degree in marketing and a master's degree in computer science?
With that combination, you'd be an ideal candidate for a data science role. If you get into the right company, it can also be a very rewarding career with lots of interesting challenges to solve. It can be difficult for companies to find ...
What is the ideal career for a marketing + computer science post-graduate?
Pretty much the same as for everyone else.
Basically a high paying job, with interesting and fun work, which recognises you as a hero.
Is it worth it?
All formal certifications that are internationally recognised are good to have and can be big boost to career.