I've always seen the institution as merely a fabrication in Buffy's mind, not as an alternate dimension or anything of the sort.

As for the parting shot, I think it's a beautiful allegory of Buffy finally 'giving up' on that normal life she's always struggled to have. The scenario, as imagined by Buffy, is somewhat of a perfect life for her, maybe something that was always at the back of her mind as possibility. If you look at the cinematography of the final shot, it's one of an outsider (Sunnydale Buffy) looking in, not of one (Institution Buffy) looking out.

Or on a more ominous note, it could signify the death of happy-go-lucky Buffy... Gone.