Can a sinus infection cause you to get cold sores and dizzyness and feel hot then cold?

Yes and no. You generally get cold sores when your immune system is down. A sinus infection can tax your immune system, . So they are an indirect cause.
Yes, maybe. You can get cold sores anytime but it seems like people get them when their immune system is compromised. If you get cold sores, you probably already know what conditions cause them with you. If you have a sinus infection, get it treated regardless! take care of yourself.