In short, the answer (regarding diseases) is yes.
Anytime you deal with blood or other bodily fluid, you risk catching a variety of diseases. The risk is compounded if you are not wearing proper protective equipment, washing your hands after handling them and covering breaks in skin before handling. Here is a list of the diseases you could end up with as well as a list of other fluids and how exposures can occur.
If you would like to learn more and be able to better assist your patients, you should take first aid classes. Also, if you think that you have a high likelihood of coming in contact with blood or other fluids, you should consider carrying a pair of disposable medical gloves and safety glasses in your vehicle (or backpack... Something that is often with you).
If you live in an area that has fairly prompt ambulance response, you are much better off not trying to take the patients to the hospital. Instead, if it is safe to do so, try to redirect traffic around the patient so that they don't get hit again and try to keep bystanders a few feet back from the incident so that they do not get hit as well. Try to avoid completely stopping traffic as this can effect the emergency vehicle's ability to get to the scene.
Your safety on scene is your priority, if you have the equipment, training to do so and ability to not get hit in the process, you should attempt to provide care to the patients. If any of those factors is not possible, you are much better off trying to keep the patient calm, keep bystanders at a distance - if possible - and try to keep the patient(s) from getting hit again.
I hope this is helpful to you. Please feel free to ask questions if you need clarification.