I think this is a yes and no answer.
Yes: chronic and genetic diseases can still occur
This is conjecture, as the details on how humans are 'grown' is scant (see this page). Assuming that 'grown' means that rather than through natural reproductive methods a technology such as IVF is used to artificially grown humans, errors in the genetic construction of humans would occur, leading to genetic diseases. As there is a strong link between one's 'real' physical self and one's residual image, it is expected that if one had a genetic defect, one would probably show signs of that illness within the matrix.
Regarding chronic diseases without a genetic basis, one cannot deny the power of the mind within the matrix. If people live lifestyles that would in reality lead to developing a chronic disease, then it is probable that they would indeed develop that disease. Remember that Smith said that this version of the Matrix included suffering because it is essential to humans to accept the matrix as reality. On that basis, it is highly likely that one's actions have consequences so, with that being said, the development of chronic disease from actions within the matrix probably would have consequences within the matrix as well. Getting a little sidetracked from the original question I know, but I thought I'd add it in!
No: infectious diseases probably cannot occur
Infectious diseases, in terms of a real disease probably cannot occur. Consider the following extract from the Matrix Wiki page on Earth:
Life has proliferated over billions of years, until the sky darkening
operation in attempt to block the energy source of the Machines.
Eventually almost all life became extinct, resulting in a mass
extinction bigger than the one that occurred during the Permian.
Now, this is conjecture, but if most of life has been eradicated, then the possibility of contracting an infectious disease would be pretty low, as bacteria need something to live on and it seems that Earth mightn't have the capacity to sustain such life anymore. Furthermore, the capacity to transmit diseases from one human to another in the real world would be severely limited by their isolation. As pointed out in the question, it is possible that someone with an infectious disease who is used for food for other humans could transmit such a disease. It is not unreasonable, however, to expect that there would be some filtering system, as it would be to the machine's disadvantage to foster an environment where infectious diseases could be spread.
The above addresses your first two questions as yes.
On the third question as to whether medical treatment within the Matrix would have any impact upon the physical body, I see two solutions. Either:
- It would simply be a placebo effect; the person would either get better by themselves or not, or (more likely)
- Medical treatment would be provided in the Matrix in coordination with some real world intervention by the machines.
Treatment for infectious diseases would be possibly effective, depending on what was administered, but a placebo effect would be more likely to be used to get the body to fight it off by itself. On the other hand, if the body hasn't been exposed to many antigens then the immune system would probably be abnormally weak by our standards so some external intervention would probably be required. In the case of chronic and genetic diseases, treatment options (to the best of our current knowledge) are far more limited, so there mightn't be as much to work with there.
Considering that damage to the body within the Matrix causes damage to the body outside the Matrix, we shouldn't dismiss the placebo effect. It is highly plausible that the effects work in reverse as well i.e. if one believes completely that one is receiving the best treatment possible for an illness, the mind may make it so the body repairs itself completely.