The Inspector (Edward Norton) is a biological son of Gustave (Ralph Fiennes).
It's obvious from when Norton and Fiennes first meet in the film that the Inspector holds some reverence for Gustave, considering his childhood memories of his time at the hotel. He brings up his parents directly in front of Gustave, Zero and the assembled soldiers.
Gustave asked only about the inspector's mother and not his father and he has been known to sleep with women even if they are married, as his alibi for the nigh of Madame D's murder was a married Duchess. Even if this is not true, this is now this troper's head cannon. It would also explain how Gustave remembered a child from decades earlier and how the inspector remembered Gustave.
If nothing else, Henckels' mother cheating on her husband with Gustave seems quite likely.
Bill Murray's character is Gustave's biological father.
He dropped everything to save Gustave. And I wouldn't be surprised given Gustave's own background.
Gustave was, officially, executed for being gay.
Thirties fascist regimes tended to have a severe problem with homosexuality, to put it mildly, and making sure that Gustave was executed on those grounds would have pleased Dmitri not just by humiliating Gustave, but by making it more likely that his relationship with Madame D would be remembered as purely a scam and with no genuine emotion.
It's possible Gustave H. is bisexual, as Dmitri says, and we simply don't meet any of the male friends he's slept with — at any rate, it wouldn't change the nature of it as an accusation.
Word of Godconfirms that Gustave is, indeed, bisexual. (Click on "Look Inside" and scroll down to the introduction by Hugo Guinness.)
Henkels was replaced (and possibly executed) by the fascists
Firstly because it is difficult to believe such a scrupulous policeman would be tolerated by the new regime, and secondly because of the scandal he created by revealing the truth about Madame D's will.
The ripping-up of his official note strongly suggests at the very least that there was some kind of regime change (from bad to worse).
The answer can be found in this newspaper article◊: Henckels left his position and the country to become Secretary General of the Government-in-Exile of Zubrowka during its occupation, returning afterwards to resume his functions as an inspector (and decorate Zero).
There are certainly some distrubing similarities and the ages would match up.
It would explain Chigurh's weird accent - it's actually a mix of Mexican and Zubrowkan.
Gustave is Jewish and he (or his family) fled Czarist Russia.
When Zero mentions taking an oath on a Bible (and actually produces one), Gustave quickly declares he'll do it 'later'. He's understandably stressed out having been arrested but his reluctance might have been down to being non-Christian. He looks about the right age to have been a child during the later pogroms in the Russian Empire (like the 1903 Kishinev pogram), which would also explain both his Mysterious Past and his sudden emotional rush of empathy for Zero when he learns of Zero's refugee status.