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.