Plug your phone into your PC via USB. It should say USB connected within the notification bar on the phone.

Then on the PC select start>My computer. Under portable devices double click on HTC MTP Device (or HTC One X+ if you have USB Debugging enabled under user options in phone settings), then double click on Internal storage & you should see all of your files. Copy them to your PC.

Also ensure that your calendar & contacts are synced to your Google account. To do this go to settings>Accounts& sync>Google, select the account & ensure that calendar & contacts are checked. Then from that page touch the menu (3 dots top right of screen) & press Sync now.

If your phone contacts are not google contacts you can either export to SD & back up to your PC or copy them to your google contacts. Open the People app, touch menu>Manage contacts then select either:
a) export to phone storage (virtual SD) or
b) copy contacts & copy from phone to your google account.

Having done all your back ups you might find that a Factory reset via Settings>Backup & reset>Reset phone may cure your problems but will erase user data. Might be worth trying because that's probably what htc will do first.