I agree that SpaceX and Boeing are two large employers that don’t necessary need clearance.
Yes, you can definitely work in aerospace without a degree in aerospace. In fact, most of the people I’ve worked with in aerospace DO NOT have a degree in it. My degree is in civil structural engineering and I have all kinds of hardware that I worked on in either low earth orbit or roving around Mars.
As far as having to be a US citizen; no, the requirement is more like you need to have the proper visa/ green card/ whatever to be able to legally work in this country. The aerospace companies I’ve been associated with would not sponsor a foreign citizen to get that authorization. So, if you are in the US on a student visa only, it going to be hard to find work as you need to also find that sponsorship.
Security clearances are not required for the majority of aerospace jobs. ITAR restrictions do however seem to dominate the field. But that just requires that you be a US “person” not a citizen.
I’ve been continuously employed in avaition for over 30 years as an ME.
Straight out of college, the MEs were more likely to be assigned to propulsion and structures (with the Aeros assigned to S&C, or performance). Later the engineers drifted to their (passions) skill set. My specialty was flight test conduct.