Can a pilonidal cyst cause low platelet and neutrophil count? Mass production of antibodies also attacking my platelets? Similar to what HIV does?

Pilonidal cyst. A pilonidal cyst should not cause a drop in platelets or a drop in the neutrophil count. If that occurred, I would look for another disease process that could account for that. Pilonidal cysts can become infected - and if someone is very ill from that, you might see derangements in blood cell parameters. Those infections involve bacteria, and therefore are very different from HIV related effects.
Doubtful. The cyst it's self would not cause any lab abnormalities. If it got infected you would expect it to raise your WBC's, not lower them.