What is this Google Chrome icon appearing in the dock?
Yes, the icon you see on the right is Handoff from another computer that you are signed into and are near. That computer has a window open in a web browser. (I'm assuming you have Chrome as your default browser)
You can ignore it if you want, or select it to &quot;hand off&quot; what you are doing on the other computer to this current one.
To answer your questions;

The icon that you see there usually shows what device is &quot;handing off&quot; from. In this case, it's a desktop, not a mobile device.
Unless someone is using your computer without you knowing, it's probably not a security issue. This only is shared between computers that are signed in with the same iCloud account.
Someone can Airdrop something to you, but you have to agree to it. Airdropping from a computer that are are signed into with your iCloud account to another computer, also signed in with the same iCloud account, can just transfer files without asking. No third party can just send, or push, something to you without you agreeing to it.

I hope this helps!