All Rust streams with the Drops Enabled tag will count towards earning drops. However, streamer-specific drops which will require you to watch the specific streamer in order to get their drop. All participating streamers are listed above and all active campaigns and their conditions can always be seen on this Twitch page.
You can track your progress on the Twitch Drops Inventory page.
No, you will only be eligible to receive a drop for one active channel at a time. Watching more than one channel simultaneously will not result in more drops.
It can take up to 10 minutes after pressing the Claim button for items to show up in your Steam inventory. If the item does not show up after 30 minutes then your Steam account may not have been linked to your Twitch account when you claimed the drop. You can link your accounts on the top of this page and then click the button to check for missing drops to attempt the reward again. You will need to own Rust on Steam in order to receive the items - family shared accounts will not receive them.
You can unlink your old Twitch account by signing in with Steam on this page and then clicking the Unlink Account button. Then you will be able to link your accounts normally again.
Yes, as long as you have claimed the drops you will be able to get them when you purchase Rust later. However, they will not automatically appear in your Steam inventory. You will need to make sure your account is linked on this page and then click the 'Check for Missing Drops' button.