No, it's not normal. Not sure of exactly what the cause will be.
iCloud setup generally is just a matter of signing into it for setup. Sometimes it gets corrupt and requires signing out and signing back in. Also, your keychain may be corrupt and causing the problem. You also mentioned something about iTunes syncing. I don't remember what you can sync with iTunes in Mavericks, but it isn't a good idea to use to different methods for syncing the same items. So no need to sync Contacts when iCloud already syncs them.
You can try this process unless you want to wait for someone else to reply. This may take a few minutes, but shouldn't do any harm.
Hopefully you do have a backup in case anything goes wrong.
If you are using iTunes for syncing, uncheck anything that might be duplicated in syncing by iCloud such as Contacts, Calendars, etc.
Quit all applications.
Go to iCloud.com and make sure all information for Contacts and Calendars is correct. If all your info is correct at the website, then go to System Preferences and open the iCloud pane. Don't uncheck the boxes in the list, just sign out of iCloud completely. Choose to delete all info such as Contacts, Calendars, etc. from the computer when asked. This should remove all the information from the application as well as the databases for those on the computer.
Also, pen the Message and FaceTime applications and sign out of those also. Now you should be completely signed out of all iCloud applications.
Next, open Keychain, and select Logins in the top window and All Items in the bottom window. Search for iCloud and delete any remaining items related to iCloud logins. I'm not sure if Mavericks uses or if you are using the iCloud keychain. Note what keychain an item belongs to. Only delete from the Login KeyChain and not an iCloud keychain. If you do, you may need to re enter the passwords on the other devices when asked.
Then open KeyChain first aid and verify and repair your Keychain.
Open Disk Utility and select your boot volume from your hard drive. Repair permissions.
Restart the computer. I assume you should not get any requests at this point for iCloud logins.
Go to iCloud in System Preferences and sign into iCloud. Verify any service you use is checked such as Mail, Contacts, Calendars, etc., and leave any you don't use unchecked. Verify your Contacts, Calendars, etc., are restored.
And last, go to Messages and FaceTime if you use them and sign back into those also. Make sure all your addresses in each application are shown. You will probably get alerts on other devices that all those addresses have been signed into by another device. That is normal.
If this doesn't solve your problem, then I don't know what else I can tell you.