Can uncircumcised penis bleed after having sex without a condom?

Yes. Penile bleeding can occur after any type of trauma (sexual or otherwise). A condom can protect you from infections causes but not traumatic ones. If you are uncircumcised, you may have penile adhesion or fissures that cause bleeding. Regardless, seek an opinion from your provider if it persists or if you are concerned about an std.
Shouldn't. If you have penis bleeding after sex, you're doing something wrong. Also, why are you having unprotected sex. Haven't you hear of std's? Check w/your doctor.