Can you cause the anus to bleed a bit if rubbing a bit too hard with cheap rough toilet paper?

Yes: Rubbing too hard or using rough toilet paper can cause anal irritation and cause slight bleeding. You can prevent this by NOT rubbing too hard or using softer toilet paper. More importantly, are you suffering from anal itchiness? If so, you need to see your primary physician for evaluation, possible treatment and/or referral to a specialist.