The "about" section tells you the version in the title bar, like AnkiDroid v2.0. If you got it from the Play Store recently, you likely have version 2.
The message you get about syncing is something that happens when you make certain changes to your collection, like adding a note type. It's a regular occurrence in Anki when modifying your collection. It's not a problem. If you've been moving the files manually without syncing, then the reason the message comes up is because AnkiWeb doesn't know about the changes you've made.
Since your PC has the latest version you want to keep, choose the upload option from your PC when syncing. AnkiDroid will now get the same message when trying to sync. Since you want AnkiDroid to have the new changes as well, choose the download option from AnkiDroid when the message comes up.