All Rust streams with the Drops Enabled tag will count towards earning drops.
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 have Rust in your Steam library to receive the items.