Right when you first load an image in PS through LR using Edit In it loads the RAW file with its original file extension. But the second you SAVE that file it saves it with the extension, and type, of what you have chosen in the External Editor section of the LR preferences. In your case with the TIFF extension.
This is completely normal. Also when you send a file to PS using Edit In you do not get a Adobe Camera Raw window because LR "IS" Adobe Camera Raw in a different looking package.
Ok, so the delay in opening is mostly LR doing a RAW to TIFF conversion if I understand correctly. Sounds fine, a bit silly that the filename extensions don't change though.
What about PS opening two files side by side, one labelled .nef and one .tif?
The title of a document in PS uses the extension of whatever the file was opened from or saved to if you’ve already saved it since opening. Since PS is actually opening the raw file it uses the raw file’s extension in the document title, specifically NEF in your situation.