Allows as in initiating a cascade of biochemical reactions leading to SARS-CoV-2 cell infection.

If I'm not wrong, any cells with ACE2 and TMPRS22 expression can be infected by SARS-CoV-2. Yes, SARS-CoV-2's genes reprogram the cells to make its spike proteins and other virus components, which re-assemble to form the mature virions that exit the cell to infect other cells. The similar concept applies to vaccines, except that only the modified spike proteins get produced by the cells.

The immune system reacts to the Covid-19 vaccine in various ways, in which the most important ones are the antibody and T-cell responses. This is not autoimmunity since the vaccines are supposed to work this way. Autoimmunity means that the immune system made a mistake in targeting cells it does not mean to.

I'm not sure what do you mean by 'program', 'entity,' or 'entry vehicle'. If you mean other components of the vaccine such as lipid nanoparticles or other adjuvants, then the answer is no, as far as I know. Unlike viruses, vaccines (except live ones) don't replicate themselves.

While immunocompromised persons may not be able to neutrlaize the vaccine-induced spike proteins completely, it should not be a problem. Vaccine-induced spike proteins don't last very long (i.e., about one week), based on animal studies. I'm not sure if this applies to every single case, so outliers may exist. At the same time, however, immunocompromised patients are at a much higher risk of severe and fatal Covid-19, so the risk-benefit analysis needs to be considered more carefully.