are there any broad clues to reflexive verbs, since they are so much more numerous in French than in English, or is it just raw memorization, like gender and irregular verbs?
By "clues", do you mean a way to tell which English verbs are reflexive in French? Not really. Perhaps if you think of reciprocal or reflexive actions in English, you'd get somewhere, but no guarantees. For example, "se téléphoner" (to call each other) and "s'asseoir" (to sit yourself down) would be fairly obvious reflexive verbs.
The verb "se méfier de" is a particular instance of an "idiomatic" reflexive verb, because the word "méfier" only exists in this reflexive form and its meaning is idiomatic, as opposed to reflexive or reciprocal.
If your question is about grammar, then studying the rule for reflexives is quite an extensive topic, but there are plenty of resources online to get you going.
TFW your native language has no reflexive verbs and you have no intuition for such idioms. Each and every reflexive verb is a complete surprise and basically none of them make sense to you.
Eg. You mention "to sit yourself down" as being fairly obvious, yet it too seems woefully complex to me (and by no means obvious or expected) - we have separate verbs for "to sit", "to sit down", "to take a seat", "to seat someone [somewhere]" etc., all expressed by a single word.
(We do have reflexive pronouns, but you only really use them when you specifically want to highlight that the subject[s] are doing something to themselves - and that this something to note - but this is a stylistic choice, not something grammatically mandated.)
Darned linguistic intuitions! At least I can use English to make some sense of French (and other Germanic/Romance languages), studying using my native language would be a rather significant hindrance.
To turn this into something else than just a distracting sidenote 'rant' and provide at least a smidgeon of potential value for anyone who might've read through this - might I ask if you know any good quality sources on reflexive verbs?
- Elle se méfie de son médecin à lui.
The subject will give you an obvious clue to which possessive is used, but using "his" is technically still valid, you'd just have to clarify in a real conversation. I don't guarantee that Duolingo will accept both all the time.