I had an angiogram three weeks ago. Except for a minor pricking from the injection of painkiller at the top of the leg I felt nothing whatever, watched the arteries moving on the screen and talked to the doctor while he was maneuvring around them.. As it happened, a stent was put in but again, I felt nothing. I was in truth a bit uncomfortable afterwards when the anesthetic (which is only given to avoid discomfort at the entry point) wore off but I asked for a painkiller and calmed down. Go ahead. Have no fear.
You could ask for a ct angiogram if you are worried, which is basically a non invasive scan.
However, to answer your questions, the angiogram is very safe as far as invasive procedures go. The biggest risks are stroke and heart attack, but for this to happen the cardiologist would basically have to disturb a chunk of blockage material in the arteries with his catheter wire.
It isn't really about odds, are you the likely or unlikely odd? It's about your symptoms and ruling out cardiac issues. As far as I'm aware, there are benefits to either a CT or standard angiogram. With a CT it picks up more vessels with greater detail, but just one image, giving a static view of your heart. In other words, you don't see the heart and arteries working. An angiogram is actually real time, you see the arteries with the heart beating. If you suffer from artery spasm then this is more likely to pick it up visually especially with a catheter in the vessel. The other advantage is that if they see a large blockage, then they could decide to stent it there and then, something which cannot be done obviously with a ct scan.