The critical point, alluded to by Azula R in the comments, is that the risk reflects incident cases, while prevalence reflects both new and existing cases. Thus prevalance is a function of incidence and duration of disease. This distinction is important when trying to identify potentially modifiable causes of disease. If we identify a variable that is strongly associated with increased prevalance, intervening on that variable might do nothing to prevent future cases, because it might only be related to disease duration.
First, the incidence rate is defined as the number of new cases over a time period divided by the total person-time at risk during that time period. I'm assuming you are interested in the incidence of disease and not the incidence of death from disease. So, you don't need to worry about the 3000 deaths from disease in your incidence rate calculations, because these are neither new cases (if they died from the disease they must already have gotten the disease and therefore be included in either the 5000 new cases or the 20000 existing cases), nor do they contribute any person-time at risk, once they have the disease.
In your example, we do not have the exact about of person-time that each individual contributes, and we know that there are differences in the amount of person-time contributed, because some people are dying. We can instead calculate the cumulative incidence and convert that into an estimate of the incidence rate using the following formula: $CI = 1-e^{-I*t}$ where CI is the cumulative incidence, I is the incidence rate, and t is the time period. Cumulative incidence is calculated as the total number of new cases over the time period divided by the number of people at risk at baseline: here there were 5000 new cases and 80,000 (i.e. 100,000-20,000) people at risk on Jan 1. Your time period is 1 year. Note that this calculation assumes an exponential distribution of incidence times, but since you are given no information about failure times it is a reasonable assumption here.
The other questions ask about prevalence and this is where you may need to subtract the people who died. Prevalence is calculated as total number of cases existing in a population at a given time divided by total population at that time. So, on Jan 1 you have 20,000 cases out of 100,000 people. On Dec 31, you have (20,000+5,000 - 3,000) cases and 100,000 - 3,000 people, because 3,000 people have died. Since they died from the disease, we subtract them from both the total number of cases and the total population. If they had died without ever having the disease, we would only subtract them from the total population.
Best Answer
Incidence refers to new cases of a disease, while prevalence refers to all existing cases.
The two are related (Prevalence = Incidence x Disease Duration), but aren't the same thing. For example, for an incurable disease like HIV, the incidence rate could be decreasing as prevention measures improve, but the period prevalence can still rise (due perhaps to increased survival rates).