Is it normal that I am scared to poop after giving birth?

Yes. Your intestines take a while to get back to normal, so it may take several days for you to need to have a bowel movement. In that time, the stool can get hard and be difficult to pass. It's a good idea to take a tool softener until things are working right. And keep up your fluids!
Yes. Many women are fearful they will tear out any stitches with a bowel movement after delivery. It won't happen, but taking stool softeners to keep your stool soft will decrease the need to strain and cause pain.
Yes. Many women are concerned about this. If you feel like this will inhibit you, you can give yourself a quick enema at home before heading off to the hospital.
Yes. It is exceedingly normal. You have suffered a very traumatic event to your pelvic floor (even though you received a wonderful present for doing so) and there is a natural fear about doing further damage or reopening freshly repaired tears by going to the bathroom. This is exceedingly unlikely to happen, even if your tear went into the rectum itself. It would be recommended to use stool softeners.