Guests cannot really use a forum where only the topic author can view their own posts, because there is no "author" for the posts. We don't have any way to know if a guest makes a post and comes back a week later that it is the same guest, so they're inherently unable to use such forums.
You don't need the "Hide last post info" setting because the software automatically figures out what the last post the user can view is. In 3.x we cached the post data to the forum and showed it without checking permissions, necessitating that setting - that's not the case in 4.x.
You can perform a search on the members area without a keyword.
https://invisioncommunity.com/search/?&type=core_members
This is a widget that can be added using the sidebar block manager.
AdminCP > Members > Moderators and AdminCP > Members > Administrators. You add the members or groups you wish to give access, and then set their restrictions (if appropriate).
You "hide" content that has already been approved. You cannot "unapprove" it, but these are just semantics really.
If you modified the templates, you could do this. Announcements are not designed to look like topics in 4.x out of the box.
This can be configured in the aforementioned AdminCP > Members > Moderators.
Not out of the box I'm afraid.
Yes, in AdminCP > Customization > Editor Toolbars you can double click on the "link" button and control which groups have permission to use it,
BBCode is technically possible, but you're misunderstanding how the system works. With a WYSIWYG editor, we still filter and clean the content on the backend using a well-known and well-tested HTML filtering library. You can use the WYSIWYG editor without being concerned about security issues.
We removed the "avatar gallery" feature in 4.x so no.
AdminCP > Customization > Icons & Logos
Yes, in the file storage handlers in the AdminCP you can specify the path to store the files, and then specify a custom URL (to facilitate using a CDN to serve files from that path).
But please keep in mind, you are posting in a feedback forum designed to allow clients to suggest new features and ideas. What you are after here is technical support and "how do I" stuff, which should be posted in one of our other forums, which you'll have access to when you renew your license. You're also welcome to contact sales@invisionpower.com if you have any other questions you'd like answered before purchasing a license, and/or you can sign up for a demo of 4.x if you need to on our main site.