There are little veins inside our noses. If we blow our nose continuously it is normal that blood appears.
Have you ever had blood coming out of your nose of suddenly during summer or when it is really hot out?
Our veins dilate and then blood comes out.
So when we blow so often our veins also suffer and then they can bleed.
But it's very normal. Virus don't cause colds. Bacteria does. Any of these won't make things worse if you bleed.
If you have a cold or a virus infection it's because you have this virus or bacteria already inside your body.
Remember that your body has agents to fight against external agents, and you will survive