Why not? Just because it doesn't match up, or is there a deeper reason?
Reverse transcription only happens when a reverse transcriptase (found in retro-viruses, e.g. HIV-1, HIV-2, Hepatitis B Virus etc) does that and the RNA is the direct viral genome or a transcript of the viral DNA (for DNA viruses). They have special marker sequences at either end to facilitate this (search for terminal repeats)
SARS-CoV-2 is not a retrovirus.
eukaryotes use splicing mechanisms that copy/delete/replicate DNA from one locus to another, this is more reliable as mRNA are unstable and become damaged quickly. cytoplasmic mRNA is not of fidelity and does not reverse compile into the dna sourcecode, or it would produce massive errors and corrupt the DNA.
The CDC and other sources argue the vaccine is safe because it doesn't alter your DNA, but I have a vivid imagination and can think of a thousand other ways something might go wrong. Having my DNA altered was never one of my concerns.
what you would get is some false starts, and segment of protien if that at all.
Firstly they only express it as long as the mRNA remains intact. mRNA are degraded naturally so there is a finite time window as the mRNA is not reverse transcribed into DNA and reintegrated into the genome.
> what are the chances that it will also learn to react to other proteins expressed by those cells,
T (and B cells? I can't remember) cells undergo a selection process in the thymus that select against cells with receptors that bind to self proteins, that is proteins made normally by the body. Sometimes this fails and you get autoimmune disorders. Chances are very slim and no more than any other antiviral vaccine.