If someone has lymphoma and gets sick. Will the cancerous lymph node hurt like a normal one would as the result of the infection?

Probably not. I think the question is would an enlarged lymph node caused by an infection hurt as compared to a lymph node enlarged with lymphoma. Generally, an infllammed or infected lymph node from infection, hurts. A lymph node enlarged by cancer cells, such as lymphoma, is not tender and does not hurt. That's why self examination for enlarged nodes (axillae in females, groin in males, eg) is so important.
Not necessarily. Enlarged nodes due to lymphoma or other forms of cancer may be somewhat tender, but usually not to the degree that inflamed nodes from infection are. Often the consistency (the feel of these on exam) is also different from infection, but the best way to tell the difference is by biopsy.