Do I have cold or sinus infection? Runny/stuffy nose, itchy scratchy throat, top right teeth seem sore, and my front right tooth has become super sensitive to temperature over night, now I seem to be feeling very cold, a dry cough and very chapped lips

Maybe both. But usually when sinuses get infected the upper teeth become sore, the infection in the sinus can creat pressure which can push against the root of the teeth hence the sensitivity and soreness. The rest of your question i hope an md can answer! hope you feel better soon!
Very possible. Sinus infections can cause the upper teeth to hurt, especially the back ones. Combined with your other symptoms i would think you have a rather bad cold combined with a sinus infection. I would visit your doctor asap for an evaluation.
Possibly. Sinus infections often cause upper teeth, especially molars, to become painfully and hypersensitive because the roots of these teeth and the nerves that exit them are in close proximity with the sinus floor. Sinus infections sometimes change your bite makeing the infected side hit first when you close since the bone underneath the teeth can swell. See a dentist to rule out dental pathology .