Here's a self-answer addressing four things:
- Actually, it turns out you don't have to use dative pronouns. You can also use possessive pronouns.
- When the dative pronouns are used, they are not ethical datives.
- How one describes pain in a body part.
- Whether one always needs to identify the owner of the body part, as in English.
It turns out that you can, after all, use a possessive pronoun to identify whose body part it is. You can also use the dative. I haven't seen any case where you use the genitive, so I'm thinking that that isn't idiomatic. Here are two examples from Homer:
(a) Unambiguously a possessive pronoun:
Ζεῦ ἄνα δὸς τίσασθαι ὅ με πρότερος κάκ’ ἔοργε, δῖον Ἀλέξανδρον, καὶ ἐμῇς ὑπὸ χερσὶ δάμασσον
Here ἐμῇς is feminine plural, agreeing with χερσὶ, so it's definitely a possessive pronoun. (It also happens to be dative, but that's just because χερσὶ is dative.)
(b) Unambiguously a dative pronoun:
αἶψά τοι αἷμα κελαινὸν ἐρωήσει περὶ δουρί.
In this example, τοι can't be a possessive pronoun, because it doesn't agree in case with αἷμα; τοι is dative, αἷμα nominative.
These are what Smyth refers to as the dative of the possessor (p. 341, sec 1480) and Benner calls a dative of interest. Benner says:
The dative of interest ... is often found where the English idiom requires a possessive adjective or noun.
He then gives the examples of Iliad 1.55, 1.104, and 1.188, which all involve body parts.
When a noun is modified by a possessor and the noun is in the dative, the possessor is in the genitive, and vice versa. This seems to be the only situation I've come across in which the possessor of the body part is genitive:
ὅς κεν ἐπ᾽ ἤματι τῷδε πέσῃ μετὰ ποσσὶ γυναικὸς (Iliad 19.110)
Re #2: These are not ethical datives. From the grammatical descriptions I've seen, an ethical dative is an indirect object of a verb, or, in Smyth's description, it modifies the sentence as a whole (p. 341, sec 1474; p. 343, sec. 1486.). But in the following examples the dative pronoun modifies a noun (p. 346, sec. 1502):
ὄσσε δέ οἱ πυρὶ ἐΐκτην
αἶψά τοι αἷμα κελαινὸν ἐρωήσει περὶ δουρί.
In these examples, the dative pronoun modifies a noun in a manner similar to the more common one in which a genitive noun modifies a noun. In the first example, ὄσσε is in the nominative dual and is modified by the dative pronoun οἱ. In the second example, αἷμα is in the nominative and is modified by the dative pronoun τοι.
In neither of these examples is the dative pronoun acting as an indirect object of the verb or to modify an entire sentence, as in Smyth's description of the ethical dative. Furthermore, the ethical dative is "used to denote the interest of the speaker, or to secure the interest of the person spoken to, in an action or statement" (not the case in 1.104) and "The dative in the third person is very rare" (but the third person is used in 1.104).
These pronouns are also not possessive pronouns, which would agree with the noun they're modifying in gender and number.
These constructions are also not analogous to ones in Romance languages such as French "Je me lave les mains" (fdb's example). In this French sentence, "me" is a reflexive pronoun that acts as the indirect object of the verb. Although, as an indirect object pronoun, it can be said to be in something analogous to the dative case in earlier IE languages, it is not modifying "mains," and therefore is not analogous to constructions like the ones in Iliad 1.104 and 1.303.
Cf. Smyth p. 351, sec 1531, re the use of the body part in the dative.
Re #3, pain:-- In English we say, "My foot hurts," but there is no intransitive verb like "hurt" in ancient Greek. There seem to be a variety of ways of expressing this kind of thing in the Homeric dialect, but the most common is expressions involving πάσχω, to experience or to suffer, with the dative used for the body part:
πάθοι ἄλγεα θυμῷ -- suffer pain in one's soul
The noun πῆμα is also used, as well as the plurals.
Another common idiom is that the injury or weapon weighs down or oppresses the body part. I don't know if this is poetic or is something that people would really say.
Εὐρύπυλος δ᾽ ἐπόρουσε καὶ αἴνυτο τεύχε᾽ ἀπ᾽ ὤμων.
τὸν δ᾽ ὡς οὖν ἐνόησεν Ἀλέξανδρος θεοειδὴς
τεύχε᾽ ἀπαινύμενον Ἀπισάονος, αὐτίκα τόξον
ἕλκετ᾽ ἐπ᾽ Εὐρυπύλῳ, καί μιν βάλε μηρὸν ὀϊστῷ
δεξιόν· ἐκλάσθη δὲ δόναξ, ἐβάρυνε δὲ μηρόν. (Iliad 11.580)
We can also talk about enduring or being weighed down by pain using τλάω, or we can say that pain seizes us:
τλῆ δ’ Ἥρη, ὅτε μιν κρατερὸς πάϊς Ἀμφιτρύωνος
δεξιτερὸν κατὰ μαζὸν ὀϊστῷ τριγλώχινι
βεβλήκει· τότε καί μιν ἀνήκεστον λάβεν ἄλγος.
So to summarize, the idioms for pain seem to be:
- I suffer pain -- using τλάω, πάσχω, ἄλγος/ἄλγεα, πῆμα/πήματα, and optionally the dative for the body part; example: "πάθοι ἄλγεα θυμῷ"
- Pain or a weapon weighs down my body part -- using βαρύνω + acc (note semantic similarity between τλάω, βαρύνω)
- Pain seizes me -- using λαμβάνω
These examples also show that we can talk about body parts without explicitly referring to whose body parts they are. For example, "ἐβάρυνε δὲ μηρόν" would translate as "and weighed down [the] thigh," whereas in English we'd have to say "and weighed down his thigh."
TKR has also given some nice examples in comments (quoting):
4.123 νευρὴν μέν μαζῷ πέλασεν "he brought the bowstring close to his chest"
5.345 χαλκὸν ἐνὶ στήθεσσι βαλών "throwing a bronze [spear] into his chest"