Go to iCloud.com, login with your Apple ID, go to the contacts area, and see if everything is there. If so, you can then export all your contacts, add more, etc.
Yes, your contacts are automatically backed up when you sync your iPhone. They are stored with the rest of your iPhone's data backup under the control of iTunes.
I would do both. I use the cloud but also about once every few weeks back up my iPhone 5 on my computer! If what is in your phone is important to you, it does not hurt to do both.
Yes, your contacts are automatically backed up when you sync your iPhone. They are stored with the rest of your iPhone's data backup under the control of iTunes.