As per the Trademark Guidelines
So for Twitch’s emotes, it would be for them to grant you express permission. It’s not for me to say if submitting an extension for review, and it passing that review, would count as permission or not. If you were to use assets that Twitch owns the copyright to where you need express permission you would certainly have to make it clear what you are using and how you are using it.
As for Partner/Affiliate emotes, which the copyright is owned by the partner/affiliate themselves (and they license its use on Twitch), it would be up to them, as well as Twitch, to decide if you can use their emotes. This could potentially be something that you could have the user agree to in a ToS that they agree to let your extension display their emotes (and again, you should be clear the scope of the use of the emotes, ie if you’ll be using the emotes only on their channel or if you intend to use those emotes on other channels with the extension). And you certainly can’t use emotes owned by channels who don’t use your extension unless you reach out to them and get permission first.
This subject may also get a little more strict in the months ahead due to article 13, which mainly will impact broadcasters who use content from outside of Twitch, but it’ll also impact extension devs too. Twitch has a post on that here: https://blog.twitch.tv/from-our-ceo-a-letter-to-twitch-creators-about-article-13-16ae8ec41c70
As for question 2, components can be a maximum of 50% the width of the player, and 100% of the height, and they can’t overlap. So if your extension will be full screen it has to be an overlay, not a component.