Vaccines are ways to mimic infection (hopefully) without the side effects of real infection, like getting sick.
The immune system is complex and layered. The body's primary response to an infection is to make the body inhospitable. We know this as a fever. Some infections, like the common cold, are defeated here.
There are more tools available for more resilient infections, one being antibody production. Antibodies are proteins that bind to pathogens, to act as a marker. Once marked, other mechanisms (white blood cells) can clean up.
It takes time for the immune system to design an antibody that can identify the pathogen. The produced antibodies generally hang around for a few months, or longer. This is driven by chemical factors. Fortunately, Memory T-Cells will last a lot longer, and these contain the information needed to identify the pathogen and produce new antibodies. If the same pathogen appears again, antibodies can be produced quickly.
An antigen is anything that triggers an immune response. This includes pollen, if you have hay fever. In the context of COVID, the antigens are usually spike proteins, which stick out from the surface of the virus.
It's also worth noting that this process isn't perfect. Exposure does not mean immunity. Immunity is not binary, either. T-Cells may prevent symptoms entirely, or their may be a lag.
So, now I think I can answer your questions.
The vaccine provides the antigens using a form of COVID that won't, or shouldn't, cause infection. This will still cause an immune response, which may be strong enough to cause side effects. The body will see the antigens and generate antibodies. The antigen/antibody combination is stored in memory T-cells. This process takes several weeks, and is strengthened by a second exposure.
If you already have COVID, unfortunately the vaccine won't help. Vaccines are primarily preventative. There are a few diseases with post-exposure vaccination (anthrax), but there are usually many contingencies involved. If you have had COVID, you should still get the vaccine, because you are not guaranteed immunity.
You may still be contagious or symptomatic after having the vaccine, but this should (at least) be reduced. There is still a lag between becoming infected and immune response, even if you have immunity. This will be much shorter in the immune group, however.
It's hard to say how long immunity will last for. On an individual level, you can find pessimistic cases indicating very short immunity. Remember everyone's response is unique, and unlikely cases are usually most reported. The wildcard is a mutated virus evading immunity. To eliminate COVID would be to vaccinate the world, which is highly unlikely given substantial portions of the world haven't yet admitted they have it.