Regarding the activities but remembering the desktop, please make sure that System Settings → Workspace Behavior → Activities → Switching is set to “Current Desktop: remember for each activity”.
Regarding opening links in the browser, even if it’s in another activity - this is the expected behaviour, but I’m concerned about the “activity closed” today you mentioned - did you mean that you open the activity panel and pressed “stop” on the activity where the browser is, but then opening a link didn’t open a new browser window?