Is it normal to bleed during aggressive teeth cleaning and to have sores in your gum and mouth 3 days afterwards?

Could be: By aggressive teeth cleaning, i'm assuming deep teeth & gum cleaning (scaling & root planing). Healthy gums do not bleed but if you haven't been flossing properly, regularly & you've just had a deep cleaning, it's not uncommon to have sore gums & bleeding for a couple of weeks. Hopefully, your dentist/hygienist has given you good oral hygiene instructions. If still bleeding after 2 wks, get seen.

Sometimes: Depending on how aggressive, and if all the tartar was cleaned off, it can happen. Check with your dentist, and rinse with salt water in the meantime.