What is this construct called?

[clawgrip]

It's true that that can be used, although to me it seems mainly like something you would say to yourself when you were trying to decide something, rather than as an actual question to other people.

[Trebor]

I'm interested in this phenomenon as well. How do non-Indo-European languages handle it?

In English, we also have sentences like:

(1) Am I to believe that what you're saying is entirely factual? I'm doubtful, given your long record of dishonesty.

(2) The assignment is to be handed in by 4:00 PM sharp next Friday.

(3) Our crazy professor is late again, but that's to be expected.

I find it odd to replace "to" above with anything else. Perhaps because "to" appears in what are set phrases for me.

The first sentence may not work in French, but the second and third should:

(4) ?Je suis à croire que ce que tu dis est entièrement factuel ? J'en suis douteux, vu ton long bilan de malhonnêteté.

(5) Le travail est à remettre le vendredi prochain avant ou précisément à 16h00.

(6) Notre prof fou est encore en retard, mais c'est à s'attendre.

I have no idea how these would be rendered in other languages, though.

Some such questions work for me, but are fairly idiomatic (and non-specific), such as "what to do" and "where to go".

OK, that's mostly what I have encountered.

Asking a question like

(7) Class has been cancelled at the last minute, so what to do now?

works for me, but it isn't the usual method of creating interrogative sentences in English.

Interestingly, the only case in which I can skip the "to" in such contexts is with "why":

(8) The weather is miserable today, so why bother getting out of bed?

(9) *The weather is miserable today, so why to bother getting out of bed?

As far as I know, French doesn't make such a distinction:

(10) On a annulé le cours à la dernière minute, alors quoi faire maintenant ?

(11) Le temps est épouvantable aujourd'hui, alors pourquoi vouloir sortir du lit ?

[Astraios]

I'm interested in this phenomenon as well. How do non-Indo-European languages handle it?

Hebrew does it like so:

(1) Am I to believe that what you're saying is entirely factual? I'm doubtful, given your long record of dishonesty.

Ani amur lehaamin...
1SG.NOM supposed INF-believe
"Am I supposed to believe..."
(basically the same as English, but you can't drop the 'supposed' like in English)

(2) The assignment is to be handed in by 4:00 PM sharp next Friday.

Yesh lehagish et hamesima...
EXST INF-hand.in ACC the-assignment
"There is to hand in the assignment..."
(yesh 'there is' + infinitive makes an impersonal obligatory 'one must')

(3) Our crazy professor is late again, but that's to be expected.

...aval ze haya tsafuy.
but that be.PST predictable
"...but that was predictable."
(I guess you could also use the obligatory structure in (2), aval haya letsapot laze "but there was to expect this", but it wouldn't be as natural)

(7) Class has been cancelled at the last minute, so what to do now?

...az ma osim akhshav?
so what do.PRS.PL now
"...so what are [we] doing now?"
(I guess you could use an infinitive, az ma laasot akhshav "so what to do now", but again, less natural)

(8) The weather is miserable today, so why bother getting out of bed?

...az lama bikhlal latset mehamita?
so why at.all INF-get.out from-the-bed
"...so why at all to get out of the bed?"
(this one has to be the infinitive, it sounds wrong with an impersonal like in (7) - I think because it refers to a future action where (7) is present)

[hwhatting]

I'm interested in this phenomenon as well. How do non-Indo-European languages handle it?

I'll give examples from IE languages (Russian and German), but I think they're interesting as well:

(1) Am I to believe that what you're saying is entirely factual? I'm doubtful, given your long record of dishonesty.

German uses sollen here (cognate to English "shall"):
Soll ich glauben, dass... ?

In Russian, you can render that construction by:
(1a) Я должен верить, что .... ? (Lit.: "Am I obliged to believe that ...")
(1b) Мне (можно (ли)) верить, что... ?
I-DAT (possible (INT)) believe-INF, that ...
"(Can) I believe, that ..."
But more idiomatic would be something like Ты что, думаешь, что я тебе поверю? "You, what, think that I'll believe you?" or Как я тебе поверю? "How I'll believe you?"

(2) The assignment is to be handed in by 4:00 PM sharp next Friday.

German can use either sollen or müssen (cognate to English "must")
Die Aufgabe muss / soll bis nächsten Freitag um spätestens 16 Uhr eingereicht werden.

Russian uses an impersonal conctruction with nado "(to be) necessary":
Задание надо сдавать до пятницы в 16 часов ровно.
assignment-ACC necessary hand=in-INF until Friday-GEN in 16 hour-PL.GEN smooth

(3) Our crazy professor is late again, but that's to be expected.

German is almost like English, only that "to be" is in the past tense:
Unser verrückter Professor ist mal wieder zu spät, aber das war zu erwarten.
Russian has either но это можно было ожидать "one could expect that" or, more idiomatic, но этого стоило ожидать (lit. "it was worth to expect of it")

As a sort of result, German renders English "is to" mostly by sollen or müssen, while Russian prefers various impersonal constructions, where it in general changes the subject of an English passive sentence into the object of an active sentence with an impersonal subject.