My knee area hurts considerably for a few minutes after I stand up from squatting. Why?

Knee cap. This is common if you have some wear and tear/arthritis behind your knee cap(patella). Sometimes it is just inflammation around the cartilage. Some doctors call this runner's knee, or chondromalacia of the patella. In some cases, it is frank arthritis.
It depends. That is a common symptom of patello-femoral knee pain which comes from cartilage softening or wear related to the high degree of force associated with this motion or activity. Though it would require a full medical history and exam and likely some imaging to confirm this, generally this can be improved with physical therapy exercises. Seek the evaluation of an orthopedic specialist to confirm.