I imagine See Invisibility would detect it, but since it looks like "a faint shimmering in the air that is 4 feet wide and 8 feet high" they'd have to know the spell to know what they're looking at.
I'm not 100% up with how Detect Magic interacts with invisible things, but as far as I know it will detect "something". You'd learn the location of the "door" as well as the strength of the aura ("strong"), and with a check you'd learn the school of magic. Also remember a "Strong" aura persists for up to an hour after the spell has disipated, so they could be detecting a door that isn't there anymore .
What detect magic does is allows you to know the strength and location of any magic aura’s in the area. This is of course assuming you spend three rounds concentrating on the area. A knowledge arcane roll can determine what school of magic the aura belongs to. You cannot see anything that you could not already see. So concentrating for three rounds you would know that the square with the door contains a strong aura, and if you made you knowledge arcane roll that the aura was of the conjuration school and that is all.
You would have no way of knowing that it was a doorway to an extra dimensional space. You would get the same results if someone had cast a monster summoning VII, or a greater teleport on the square and the aura was still lingering. Now if you knew or suspected that someone had cast a mages magnificent mansion and were looking for the door it would be a reasonable assumption that was what was causing the aura. But without some prior knowledge you would have no reason to suspect it was that spell.
Detect magic is not see invisibility or true seeing. All it does is tell you that there is an aura and its location that is it.