Yes! The Pfizer vaccine, for example, enrolled 45,000+ people, of whom 25,000+ people got the actual vaccine. This is on par with all other non-COVID vaccine trials, who usually enroll 30,000+ people.
Also keep in mind that other medicines (pills) are approved after only being tested in much fewer people (~1500 people, for example. Vaccines are the most rigorously tested therapies, including the COVID vaccine.
The science itself was not rushed – the only parts bypassed were the time usually required to fundraise and the time usually spent waiting in line for the papers to be reviewed by the FDA and other regulatory administrations.
No. The mRNA enters the cell but NOT the nucleus, were your DNA lives. It gives our cells instructions for how to make a harmless protein that is unique to SARS-CoV-2. It lasts just long enough to teach the cell how to encode the protein, and then the mRNA is destroyed. Then, if you’re subsequently infected, your body recognizes the virus should not be there and the immune response is triggered to fight it off effectively.
This is an amazing technology that has been in development for many years, but was able to be accelerated due to everyone’s focus during the pandemic.