It is possible, but it is a huge engineering problem. The biggest math difficulty will be the design of the control systems. Just a single jet fed from the ground is inherently unstable.
Why on Earth would you want to design and build it from scratch without the expertise to spec it? Go online and source a component to buy. I guarantee that COTS will cost less than designing and testing a new part, even if you need to have a custom interface designed by the vendor.
They did decent work for prototype quality components the handful of times my programs went that way. AFAIK, they're not suitable for military, government, or flight components. We used them for rapid turn around ground support equipment. IIRC the selection of material tempers was slightly...
Conservation of mass can't be broken. Your mass flow rate out of the combustion chamber will go down. The mass flow through the nozzle will be dependent on the fuel, operating temperatures and pressures in the combustion chamber, and the area ratios of the nozzle.
if you have a supersonic nozzle (and I'm assuming you do since there is no other reason to have a divergent portion), then you will choke the flow in the throat. Whether you are close to 2kg/s depends on the design of the rocket, though. It is much more likely that if a straight tube was...
Unless the field you're planning on going into needs a Professional Engineer licence, you might be able to find a company willing to take you on as an entry level engineer with a BS in physics.
This man speaks the truth. If you want to work in the industry, you will want a Masters unless you're designing experiments for a university or government organization.
My company (not an airline company, but heavily involved in aerospace) has had a few non US citizens working for them over the last few years.
There are restrictions which programs they can work on and where in the company facilities they're allowed to go, but they have non ITAR sensitive...
Laugh. Out. Loud.
What, exactly, did you design in your thirty years as an engineer that makes you think that a spaceship from a science fiction show using imaginary propulsion systems is the most effective method to design something?
Thinking back, my class had a team for "Design-Build-Fly" which was probably comparable to that competition. I wasn't a part of it, so I can't offer much insight to it, but it's worth looking up.
May even be able to finagle some college credits and funding from the university for a team.