do my contacts get backed up to the cloud

That depends on your settings in Settings>iCloud. Do you have Contacts turned on?

Excellent!

Technically your contacts are not backed up in iCloud. They are sync'd to all your devices that share that same iCloud account. So any changes you make to your contacts (on any of your devices sharing that iCloud account) are instantly reflected in iCloud as soon as your device connects and sync's up the changes. If you really want a backup, then use iTunes as an iTunes backup will include your contacts. That way, if you need to recover from a mistake, you can do so by restoring that backup. ICloud sync'ing will not recover information as it will always reflect the most recent changes you have made on your device - that's the difference between synchronizing information in real time versus backing it up in an archive file.

To Michael's point, this would be worth a read: About backups in iCloud and iTunes - Apple Support