The answer below draws from early buddhism material: from the four nikayas/agamas and ven. Analayo's studies on the subject (Compassion and Emptiness in Early Buddhist Meditation, CEEBM from now on).
"I'm wondering what these terms, mettā and karuṇā (loving-kindness and compassion), mean."
First, on matters of translations, while mettā is frequently translated as "loving-kindness", alternatives include "benevolence" and "amicableness"1. According to CEEBM, etymologically, it's root is mitra, "friend", thus conveying the idea of being friendly (and it favors "benevolence" over "loving kindness").
This contrasts with karuṇā/compassion which is a desire for others [in suffering] to be free from suffering. More on that below.
"Are there suttas in which they're defined, otherwise where do the definitions come from? Is there any important difference between these two terms, or do they mean the same thing?"
It seems there's no sutta that really defines compassion. The essence of it is the "concern for others to be relieved from suffering and affliction" (CEEBM). A simile that illustrates it (in the context of a teaching on how to subdue hatred) can be found in AN 5.162:
Just as when there is a sick man — in pain, seriously ill — traveling along a road, far from the next village & far from the last, unable to get the food he needs, unable to get the medicine he needs, unable to get a suitable assistant, unable to get anyone to take him to human habitation. Now suppose another person were to see him coming along the road. He would do what he could out of compassion, pity, & sympathy for the man, thinking, ‘O that this man should get the food he needs, the medicine he needs, a suitable assistant, someone to take him to human habitation. Why is that? So that he won’t fall into ruin right here.’
In terms of practice, while contemplating other people's suffering would be contemplation of dukkha, contemplating the wish for others to be free from dukkha would be contemplation of compassion and both "should be undertaken with a positive or even joyful mind [and] steer clear of sadness" instead of "commiserate to the extent of suffering along with the other" (CEEBM) -- as the Visuddhimagga formulations of "near enemy" indicate.
Furthermore, it's said in the suttas that it is "impossible, it cannot be that cruelty remains pervading the mind of one who has practiced, cultivated, and made much of the concentration of the mind by compassion" (AN 6.13). That is, cultivating the desire for others to be free from suffering eliminates the desire for others to suffer.
Ven. Analayo, in CEEBM, further clarifies that:
Moral conduct is an expression of compassion
For example, abstaining from killing beings.
Right speech is an expression of compassion
For example, evaluating if something to be said is benefical.
Both learning dhamma (seeking freedom from dukkha) and teaching dhamma can be expressions of compassion
If one does not think of harming oneself, does not think of harming others, does not think of harming both; and instead ... one thinks
of benefiting others, benefiting many people out of compassion for the affliction in the world, seeking what is meaningful and of benefit for devas and humans, seeking their ease and happiness; then in this way .. one is bright, intelligent, and with vast wisdom
-- AN 4.186 (Analayo trans.)
Also see AN 3.60 where the Buddha counter-argument a Brahmin who says that one who goes forth on the dhamma only benefits oneself.
Finally, ven. Analayo concludes that, in early buddhism, teaching dhamma is the compassionated action par excellence.
Accordingly, going into seclusion can be an expression of compassion
[ven. Mahakassapa]: "Considering two benefits, venerable sir. For myself I see a pleasant dwelling in this very life, and I have compassion for later generations, thinking, ‘May those of later generations follow my example!’ For when they hear, ‘The enlightened disciples of the Buddha were for a long time forest dwellers and spoke in praise of forest dwelling … were energetic and spoke in praise of arousing energy,’ then they will practice accordingly, and that will lead to their welfare and happiness for a long time. Considering these two benefits, venerable sir, I have long been a forest dweller ... and have spoken in praise of arousing energy."
-- SN 16.5
...and seclusion, being a necessity to build a strong foundation and practice, is then a requirement for correct teaching: "if one is not tamed oneself and wishes to tame someone else who is untamed, that is impossible" (MN 8). Also "One should first establish oneself in what is proper; then only should one instruct others." (Dhp 158)
Early buddhist discourse on compassion practice does not focus on a specific person (as elaborated on Visuddhimagga); instead, it's a practice of "boundless radiation in all directions" (MN 7).
On mettā, ven. Analayo comments on a passage of the Mettā sutta (Sn 1.8) that uses a simile of the love of a mother for her only child to illustrate the practice of mettā: "Her love of her son is not the main theme here -- the main point is protection. The providing and receiving o protection is in fact a recurrent aspect in the conception of mettā in the early discourses. [...] In contrast to the bounded love of mother for her child, which easily comes with an admixture of attachment, mettā should be boundless and free from attachment." (CEEBM).
In MN 104 we also find mettā associated with cordiality:
"Here a bhikkhu maintains bodily acts of loving-kindness both in public and in private towards his companions in the holy life. This is a principle of cordiality that creates love and respect, and conduces to cohesion, to non-dispute, to concord, and to unity."
Thus, mettā has a broad role in the daily activity establishing the modes of interactions with everyone around toward wholesome actions of body, speech and mind mind with a sense of appreciation for them (MN 31 illustrates it beautifully).
Furthermore, mettā is the very antidote for subduing (and is the opposite of) ill will, anger and aggression (MN I 424). It's also related to physical/mental beauty (and lack of it, associated to ugliness). Finally, it seems to have a proeminent role compared to other seemingly compassionate practices ("If one were to develop even just one whiff of a heart of good will, that would be more fruitful than...",AN 9.20).
So, briefly, I understand mettā to be the desire for others to be well and attitude that promotes wellness, in contrast with compassion: the desire for others to be free from dukkha -- and attitudes that extinguish existing dukkha.
"How is mettā (having desire for someone's well-being) compatible with equanimity? These Dhamma Lists warn that 'indifference' is the 'near enemy' of 'equanimity'; but could you maybe explain briefly what the right view is, how to distinguish indifference from equanimity, or to make equanimity compatible with mettā?"
In CEEBM, ven. Analayo writes: "Equanimity or equipoise, upek(k)ha, from the etymological perspective suggests a mental attitude of 'looking upon', not an indifferent 'looking away'. The term thus conveys an awareness of whatever is happening combined with mental balance and the absence of favoring or opposing" and also "equanimity is [...] a mental equipoise that rounds off a systematic opening of the heart which has been brought about through cultivation of the other three divine abodes".
As an illustration of the function of equanimity, consider teaching dhamma with the attitude of compassion to an uninterested audience. In such situations, equanimity (or lack of it) comes into play. The same could be said about the role of equanimity when we treat others with mettā and find ourselves being treated with aggression (and, conversely, when we find ourselves being praised). In this way, equanimity helps with subduing aversion and lust. Moreover, it helps putting compassion "on track", which can steer out to unwholesome mental states (by being affected by the suffering of others, or by being inflated with sense of superiority for others, etc).
The suttas have the Buddha reacting equanimous in many of these situations (see MN 137, MN 65, MN 128 and SN 7.2).
"How is mettā compatible with anatta and dukkha? Doesn't mettā imply that there are people and that they can be happy, whereas anatta and dukkha being characteristic of all compound things kind of implies the opposite?"
Other related questions I found here:
As I understand, mettā is not the idea that "other beings should be happy" (opposing dukkha), not does it reify a conceptual construction of a (everlasting) person. From mettā alone, I don't think we can infer anything, doctrinally or otherwise, about someone else (or any "object" of mettā), it's about us. It is the inclination and attitude that promotes wellness (freedom from suffering) for all beings. Like cleaning up a bathroom after using it to have it ready for the next person that need it.
The way I see it, "whishing for others to be well" is just a practice, a way to create the mental habit of such inclination and attitude -- culminating with one actually becoming a friendly, kind, "easy to be with", person.
1 from http://dictionary.tamilcube.com/
