It'd been 24 days since my last period started.... could you start feeling symptoms of an upcoming period by now? and can an upcoming period cause you to feel as though you were developing a cold because of the hormone fluctuations?

I would think that hormonal imbalance and/or an anovulatory( no egg) cycle could be causing some excess salt &fluid retention possibly simulating a head cold. Your gynecologist is best to advise you.

Irregular periods are very common and cane caused by all sorts of things. If this is recurring problem consider seeing an Obgyn. Being sick can cause a missed period. If there is a concern for pregnancy check a pregnancy test