Let's discuss your issues one by one. I will assist you with each of them.
You have attached a screenshot of Media tab where it is showing the all the media properly.
There is no button on the stream Site of BuddyPress.
Could you please explain where you are expecting the button ( I am assuming the Media button which can redirect you to the screen you have attached. )? Any screenshot will be helpful to us to understand the query.
On the Site domain.com/media (Not in BuddyPress just normal wp Page with code [rtmedia_gallery global="true"] is nothing to see (When logged in) besides "OOps...)
All files are public...
Only when logged out there is a picture of Upload and message thet only loggen in user can upload but after login it dessapears and I see again "Oops...)
If you have used this public media shortcode on any WordPress Page/Post and it is showing the photos to non-logged in users then those media will also be visible to logged-in users.
If you are talking about the
rtMedia Uploader and you want to display the uploader then you will need to change the shortcode. Please, explain to me your exact requirement I will help you to develop the shortcode.
When I have the page domain.com/media open (and see "Oops...) and than click the edit button on this site, than it's not open the media site (with [rtmedia_gallery global="true"] on it) in editor but everytime the last published blogpost. That is what I really can't explain...
I am not clear regarding your query. Please, let me know the step how I can regenerate this issue on my end so that I can help you accordingly. Also, attach some screenshot to show which edit button you are facing issue with. Have you added the Shortcode in WordPress Post/Page?
Note: To avoid any conflicts, please once try to regenerate these issues with any wordPress default theme and other plugins disabled ( except rtMedia and BuddyPress ).
I look forward to your reply.