Perfectly normal feeling.....JUST KEEP DOING IT! It only gets better :)

Absolutely normal. Especially after switching to clipless pedals!
Once you fall over at the first stop sign it's all good after that.

Riding in numbers also helps. Find even one other person to ride with.
Not only safer, but can add to your confidence as well.

Just relax, keep you eye out for crazy drivers and loose dogs and enjoy!

Ya, it can be nerve racking at times. Just stay focused on the road ahead and be conscious of your surrounding as sometimes others are not. Play it safe and stay in your comfort zone. Try to find some less traveled roads to start on to get more comfortable with the road riding and clipping in and out of the pedals. I too have gone down on my bike because I didn't clip out in time. Just takes practice, the more you ride, the more comfortable you'll get. Have fun and enjoy being out there.

What really helped my confidence on corners was to go to a big empty parking lot (on weekends...) and set up some cones.
First I just went out there and practiced my form on single corners (inside pedal up is the most important thing to know) so it became automatic. Then I worked on courses requiring me to switch directions a few times.
Finally, I decided that it would be helpful to find out just how fast I can go around those corners before I go down. Well, I did have my most painful crash (scraped elbow and ankle and massive contusion on hip from skidding) ever, but I found that I could corner _a lot_ faster than I thought possible. Now, although I of course corner a safe amount slower than tire skidding out speed, it is stilll faster than I would do otherwise....
I guess cornering is not quite as important for triathlons(I do collegiate cycling races too), but its amazing how many triathletes I've seen put there foot down at the turn a round. It is important to know how to corner confidently and without thinking when a car decides to make a right or left turn in front of you without signaling when you want to go straight. That is the most common bike-car crash. It can often be avoided by making a tight right turn with the car if you can't brake fast enough.