Do glucose levels rise if you have an infection?

Possibly. For diabetics in particular the stress and increased metabolic demands due to an infection will result in a rise in blood glucose as you convert glycogen and other stores of energy to glucose but cannot then utilize it. As a general phenomenon in non-diabetics, the same can be seen as part of the cortisol-driven stress response, which can be brought on by severe illness, including infection.
Yes. Any physiological stress can raise your blood sugar. That would include infections, trauma, burns, surgery, acute events such as heart attacks and strokes or exacerbations of chronic medical problems.