I can probably answer this better after I read the manga properly, but there's this part I happened to see in the manga that was cut from S02E06:
When Itsuki
(S02E06 spoiler)
goes to visit their visit the quints' mom's grave without the 4 older quints,
there's this part in the manga version of that
(Vol7 spoiler)
In Ch57, well at least in my translation Nino says
Itsuki is visiting the grave of 'her' (Itsuki's) mother.
Not sure what was up that weird wording by Nino...it's probably explained in the original Japanese, but Fuutarou does have this weird imagination that somehow the quints have 5 separate dead mothers who are all fathered or step-fathered by Maruo. And then the quints somehow are the same age (at least rounded to year if not rounded to month).
So idk maybe Maruo has some kind of reputation for being a playboy, not just among the nurses but even with Fuutarou...or oh wait...maybe that reputation from the nurses is what led Fuutarou to think this...oh ok I guess false alarm. I was thinking 3rd variable.
But anyway
1 -
Why did the nurses think Uesugi was the bastard son?
Right so my 2nd guess is as I said
Maruo has some kind of reputation for being a playboy
But more realistically my 1st guess is they're probably conjecturing why Maruo's footing the bill for a non-relative.
As for the sub-questions
1.1.
Is the quints' father the hospital director? If he is, is this explicit in the manga? (The anime doesn't seem to say anything directly about this.) Or is it just that Maruo paid the expenses, and he had some relation to the director, so the nurses thought he's licking the director's boots by paying his son's expenses?
Ok manga spoilers
Idk. Lol. But it says in fandom
Quote:
Maruo Nakano (中野なかの マルオ, Nakano Maruo?) is the step-father of the Nakano Quintuplets and husband of the late Rena Nakano. He runs a large hospital and he is also a doctor.
So I guess yes.
1.2.
Why did the nurse describe the director's son as bastard? Is it often the case in Japan to say bad things about your manager behind his/her back? Or is it just Maruo was bad at social, so the nurse disliked him?
Technically, the word 'bastard' was uses by Fuutarou not the nurse. The nurse could've just said 'illegitimate child' and then Fuutarou changed to 'bastard'. Of course it's possible they both said 'illegitimate child' but then your particular subtitles changed to 'bastard'.
But if you're asking or would then ask why Fuutarou was assumed illegitimate, then well...because they knew so far that Maruo had only 5 kids I guess?