1) No. A noun would take an article, and usually a plural form. Un être, des êtres for a being, beings, from the verb to be. Try that with to rain, to think, or simply the vast majority of the verbs, and it doesn’t work.

A verb becoming a noun (substantivé) is nothing more than a specific and rare etymological occurrence. They have different meanings and functions, and in general a verb and its related noun show a different form (pleuvoir and pluie, penser and pensée, etc.)

2) Yes. Subject or object, but not noun. “Souffler n’est pas jouer”, “Manger vous fera du bien”…

3) I’d say yes, but that’s what dictionaries are for. You can only be sure of those words you know.

4) There is only case b). A verb doesn’t become a noun: it stays a verb and gives a noun with its distinct life. You end up with 2 different words that only spell the same in 1 of their respective forms: infinitive for the verb and singular for the noun.

Do not confuse noun and subject. A noun can only be a subject or an object. Anything can be a subject: noun, pronoun, verb, locution, even an entire sentence.

Your question calls for a distinction between the verb in the infinitive that remains a verb when used in the grammatical place of a noun and the “infinitif substantivé”.

• The infinitive that is a full verb (what Grevisse calls infinitif pur). It is fully perceived as a verb, it cannot be preceded by a determiner or a preposition. It can hold:

  1. The place of the subject:

     Parler français est difficile.
     Être ou ne pas être, là est la question.

  2. The place of an attribute:

     Partir, c’est mourir un peu.
     Voilà qui est bien réagir !
     Cela m’était égal d’être son copain. (Camus)

     In this last example de is part of ça m’est égal (de)

  3. The place of the object:

     On entendait aller et venir dans l’enfer. (V. Hugo, quote found in Grevisse)
     J’espère partir en vacances bientôt.

  4. The place of a noun phrase :

     C’est toujours la même chose, travailler, travailler et encore travailler.

  5. The place of an adjective phrase:

     Je suis prêt à partir.

• The infinitive used as a noun (infinitif substantivé).
  The infinitive form of some verbs can have some of the attributes of the nouns, in particular they can be preceded by a determiner (always masculine), it’s the case of words like le manger, l’être, le boire, le rire, le devenir, etc.
  Among those their perception as verbs or nouns is variable. Rire will be considered as a full noun and easily be used in the plural or with an adjective:

  Oh, les joyeux rires que voici !

  Less frequent (but still used)

  Les boires et les mangers.
  Le bon manger.
  Aux dires (or “au dire”) de certains l’été sera chaud.

The infinitive used as a noun was a lot more frequent in the 13th and 14th centuries and started receding in the 15th century^*. Grevisse says Montaigne used it a lot. Grevisse adds Malherbe was probably the last French author to have made great use of it.
Later on, when used, it is perceived as an archaism. He quotes Chateaubriand:

Le passer sur les flots, le marcher sur la mousse.

^* Information found in Grevisse.

2. For example, can a present infinitive serve as subject of a sentence? ("To be or not to be, that is the question.")

.

5. How is (a) or (b) again different from de + infinitive as found, for example, in this sentence from Camus’s The Stranger

"To be or not to be" is a clause in the infinitive, not a noun. Any infinitive clause can indeed be used as a subject in French. It can appear in two ways:

In the canonical subject position, without a preposition:

[Travailler avec eux] me fait grand plaisir (formal)

[Travailler avec eux] ça me fait vraiment plaisir (colloquial)

After the verb, doubled by a dummy pronoun (cela or il formally, ça colloquially) and introduced by the preposition de:

Cela me fait grand plaisir [de travailler avec eux] (formal)

Ça me fait vraiment plaisir [de travailler avec eux] (colloquial)

(Fun fact: plaisir itself started as a infinitive used as a noun, but this is not obvious to native speakers anymore since plaisir has been replaced by plaire as the infinitive and only lives on as a substantive)

4. Do we distinguish (a) present infinitives serving as a noun and (b) nouns that only look the same as a present infinitive but are independent words in their own right?

Substantived infinitives used to, but cannot anymore, take personal or adverbial pronouns. They can’t really be followed by a predicate either, but this is less clear cut.

Compare:

(1) Ton rire m’a mis de bonne humeur (Your laugh made me happy) – Noun

(2) Rire m’a mis de bonne humeur (to laugh made me happy) – Verb

You can attach a pronoun to rire in (2), but not in (1).

(3) *Ton en rire m’a mis de bonne humeur – Gibberish !

(3) En rire m’a mis de bonne humeur – perfectly acceptable sentence

In short, a substantived infinitive acts in all ways as a noun, while an infinitive clause acts as a verb.

I’d also like to add a few titbits as a complement to the already very complete previous answers:

First is that the infinitif substantivé has been enjoying quite the comeback in the field of philosophy and psychology since the mid-19th Century, and is productively used by scholars in those disciplines.

Here a few citations (taken from BURIDANT Claude, la substantivation de l’infinitif en français in "Actes du XXVe Congrès International de Linguistique et de Philologie Romanes", 2009)

Je dis "poétique du traduire" plutôt que "poétique de la traduction pour marquer qu’il s’agit de l’activité, à travers ses produits"

Le s’entendre parler et le vouloir parler

La maîtrise, c’est précisément le "savoir-conduire" du faire.

(Notice how it contradicts what I said above about the pronouns, but this is limited to this particular usage, which sounds honestly a bit pedantic and ridiculous from outside the fields where it is used)

Another register where it is still somewhat productive is sports jargon: le toucher de balle.

Finally it is still quite common to create neologisms on the basis of le savoir-faire and le vivre ensemble: le savoir-mourir, le savoir-dire, le communiquer ensemble.

Second is that a relatively recent and immensely productive way of creating deverbal nouns of the first group verbs (those with the -er suffix) is to make their bare indicative present stem a noun:

L’entame du match (the beginning of the match, from entamer, to take the first bite of something, to begin something)

Jouer pour la gagne (play for the win, from gagner, to win)

C’est juste de la retape (It’s merely cobbled together, from retaper, to refurbish)

Elle y a été à l’aguiche (She used her charms, or more literally, "She went at it in an enticing manner", from aguicher, to entice, to seduce)

An important difference between the new and old deverbal nouns is that nouns formed on the infinitive are always masculine, while nouns formed on the indicative are always feminine.