You can edit the fav-icon by going to styles/dbt_mmo/xenforo in your forum directory. You will see favicon.ico file and you only need to replace that image to update the favicon.
how do I make it show the latest posts and topic/threads in each category:
By default you can't (as far as i know), you will need to edit the template directly to support this. Depending on system restrictions it may not even be possible (without a developer).
This is a rather large modification so i can't provide support for that, you will need to contract me at an hourly rate to apply this modification to your site. Applying this modification means you can no longer instantly apply theme updates to your site. As the updates will overwrite this custom modification, as it isn't part of the default theme. as a result they will need to be re-applied during each theme update you wish to apply to your site
Also how do I set it so they can start a conversation directly like this
You're going to need to give me a lot more context to this question, which areas do you want users to be able to start a conversation with another user?
This also may be outside of what i can do, as the conversation buttons are linked to the user account (based on client ID). So i may not be able to assist with this.