I will address your two questions in turn.
1. When can one use the indefinite article + singular count noun to make a generic reference?
As others have already said, one way to run into trouble is if, for the purpose of generic reference, you try to use a singular noun with the indefinite article in cases where you are trying to assign a property that can only be applied to a set. A typical example of such a property is going extinct: only a set of individuals can go extinct, not any one individual.
A different sort of trouble with indefinite articles is in expressions that, in principle, could have generic meaning, but which seem more likely to have a specific meaning. When you say a ring-tailed lemur lives in Madagascar, it seems more likely that you are talking about a specific (but so far indefinite) lemur. The question becomes how to judge which properties are such that the specific interpretation is more likely than a generic one. And I don't know if anyone can do better than simply give a list of such properties. Here are the well-known elements of that list: the location or existence of a type of animal, thing, or person. Other kinds of properties will be ambiguous and the context will be crucial in deciding whether the reference is specific or generic. For example, if you say a dinosaur roared very loudly, we would need more context to decide whether a dinosaur should here be interpreted generically ('any dinosaur') or specifically ('some particular, but so far indefinite, dinosaur').
- Let's assume that, in some given case, it is possible to make generic reference in more than one way (e.g. the definite article+sigular, the plural, and, possibly, the indefinite article+singular). Are there any differences in meaning between these different ways of making generic reference?
It's not really about there being any subtle difference in meaning; rather, there are some classes of things and some contexts where one way of making generic reference is much more standard than the others.
For example, it is customary in the medical profession to refer generically to parts of the body by using the definite article. So one says, This chapter deals with the lower part of the leg. It wouldn't be wrong, exactly, to write instead This chapter deals with lower parts of legs, but the fact is that no one would do that.
True, in the former case, the leg is to be understood as denoting the class of legs, while in the latter case, legs is to be understood to denote 'all the legs'. But these differences do not result in any perceptible change in the content of the final sentence. You end up saying the same thing two different ways. Nevertheless, in the context of medicine and body parts, it is so much more customary to use the definite article that using the plural sounds weird—even though it's not really wrong.
Similarly with inventions: it's the invention of the telephone; it is not wrong to say the invention of telephones, but no one would actually say that. Again, it's not that there is some subtle difference in meaning; it's just that it is customary to use the definite article when making generic reference to inventions.
These things I just said about body parts in medicine and inventions we might call 'special guidelines' for generic reference. Let's not call them 'rules', because you can disobey them and still end up with acceptable expressions, but definitely guidelines. Some other guidelines like that can be found in the excerpts I'm reproducing below.
Relevant discussions from some reputable sources
Here are the relevant sections from Collins COBUILD English Guides: Articles (pp. 36–37):
5.2 Singular count nouns with the indefinite article
You can use a singular count noun with the indefinite article to refer to something as a representative of its class.
An actor must learn to live with criticism.
An adult porpoise is six feet long.
It is always fatal to ask an expert.
You cannot use this pattern when you want to talk about the location or existence of a type of animal, thing, or person; for example, you cannot say 'A ring-tailed lemur lives in Madagascar’; you would have to say 'Ring-tailed lemurs live in Madagascar' or 'The ring-tailed lemur lives in Madagascar’. This use is common in explanations of meanings and in some dictionary definitions.
In grammar, a noun is a word which is used to refer to a person, a thing, or an abstract idea.
A mountain is bigger and higher than a hill.
As mentioned above in section 5.1, you can also use the plural without an article to express the same meaning: ‘Mountains are bigger and higher than hills’.
Note ‘Any’ sometimes has a similar but more emphatic meaning.
The greatest threat to any actor is the presumption that knowledge can be automatically transposed into experience.
5.3 Singular count nouns with the definite article
You can use the definite article with some singular count nouns to make a generalization.
The primary responsibility lies with the employer.
Here you are using one employer as the typical example of the class. This way of referring to a type of person or thing is used mainly when writers or speakers are generalizing on a topic of professional relevance to them, usually in a formal context.
This pattern is common when talking about regular participants or roles in a situation. For example, someone talking or writing about education might want to refer in general to 'the teacher', 'the learner' or 'the classroom'. Here are some examples of topics.
education the teacher, the learner/pupil, the classroom
health care the doctor, the nurse, the patient
industrial relations the employer, the employee, the shop-steward
the communication the speaker, the listener, the writer, the reader
The third task of the teacher is criticism.
In very general terms then the role of the shop-steward can be broken down into three main parts.
This pattern is often used when talking about species of animals and birds.
The red squirrel is steadily dying out.
It is also commonly used when doctors or other people are generalizing professionally about parts of the body.
This may flatten that side of the head. It won't hurt the brain.
This chapter deals with the lower part of the leg.
(See section 6.13 for another way in which the definite article is used with parts of the body.)
Inventions and technological developments are often talked about in this way.
...as a previous generation was taught to speak into the telephone.
The computer has a flexibility of function which is unique.
You can also generalize about rooms in this way.
The kitchen can be a very suitable place to practise exercise.
And here are the relevant sections from CGEL (pp. 406–407):
Generic interpretations arise with NPs that are within the scope of expressions denoting the situation type we call unlimited states. Unlimited states potentially hold for al ltime (or at least for as long as the entities which take part in them exist). Examples of clauses with referential subjects expressing unlimited states are Leo is a lion and I like Hilda. Unlimited states can be contrasted with limited states, as in Leo is angry or I am in Paris, which are understood to hold for a limited period of time unless an explicit indication is given to the contrary.
Generic interpretations are illustrated in :
 i a. Lions are ferocious beasts. b. Italians like pasta.
ii a A lion is a ferocious beast. b. An Italian likes pasta.
iii a. The lion is a ferocious beast. b. The Italians like pasta.
The denotation of bare indefinite NPs is itself unlimited in the scope of unlimited states, and so examples like [ia] are naturally interpreted generically, with lions understood as "any lion or set of lions that exists". The interpretation of singular indefinites in the same context, like a lion in [iia], is correspondingly "any lion that exists". The generic use of a singular definite like the lion is also possible in the context given in [iiia], but is an example of the restricted 'class' use of the definite article (see §8.4 below). If instead of lions, we were talking about doctors, the definite singular would not generally be possible: compare the generic Doctors are kind people with The doctor is a kind person (which only has a non-generic, referential interpretation). In The lions are ferocious beasts, the plural definite the lions would also obligatorily be interpreted non-generically. The [b] examples illustrate the generic use of nationality nouns like Italian (as well as the generic use of non-count nouns like pasta). This time the restricted class use of the definite article has a plural rather than a singular NP. The definites in [iii] could both also have a non-generic interpretation where they refer to a particular lion and a particular set of Italians.
With predicates that can only be applied to a set, a singular indefinite generic such as a lion is inadmissible:
 Lions/*A lion/The lion will soon be extinct in this part of Africa.
8.4 Restricted non-referential uses of the articles
In this section, we cover non-referential uses of NPs determined by the definite and indefinite articles that are permitted only with a restricted set of head nouns.
(a) Class uses of the definite article
NPs determined by the definite article can denote the entire class denoted by the head noun, rather than individual members or subsets within that class.
 i a. The African elephant will soon be extinct.
b. The invention of the hydrogen bomb was the next step. [singular]
c. This chapter describes the English noun phrase.
d. The human brain has fascinated me ever since I was a child.
ii a. We studied the Hittites in the final year. [plural]
b. The Greeks defeated the Persians at Issus.
The boundaries of this usage with singular nouns are somewhat indeterminate, but it is clearly facilitated in the context of species ([ia]), inventions ([ib]), and areas of study,interest, or expertise ([ic-d]). Compare, for example, *The hospital doctor is overworked with The hospital doctor is an endangered species round here, or *The tabloid newspaper is a disgrace with Hugo has turned the tabloid newspaper into a research industry. With plurals, the class use is restricted to nouns that denote nations. The interpretation of [iia] is that we studied the Hittites qua nation, rather than any particular Hittites, and similarly for [iib].