If someone asked me to describe my understanding of the english "et
cetera/etc", I'd write the following two sentences:

"I like cats, dogs, rabbits, squirrels, etc"
"Whenever I walk, run, travel, do my homework, etc, I listen to music"

In the first sentence, the "et cetera" is used in reference to nouns,
while on the second in reference to verbs.

However, here: https://www.dictionary.com/browse/etc, I read:

----------

etc.
abbreviation

and others; and so forth; and so on (used to indicate that more of the
same sort or class might have been mentioned, but for brevity have
been omitted):
You can leave your coats, umbrellas, etc., at the door.

----------

So, I don't know whether my understanding of the english "etc" was
correct, since I believed that it *could* be used too, in reference to
verbs, while here seemingly/apparently it says that it is used after a
string/sequence of nouns.

Yes, you can use etc. with a list of verbs or any other part of speech, provided the entire list is just one type (no mixing of nouns and verbs, for instance).

But the real matter is what's the case with the klingon {latlh je}; is
it to be used only with reference to nouns, or can it be used in
reference to verbs too?

If I write the following then it's obviously correct:

tlhInganpu', romuluSnganpu', vulqanganpu', latlh je vIpar.
I dislike klingons, romulans, vulcans, etc.

But can I write the following too?

jIqettaHvIS, jIvumtaHvIS, jISeDtaHvIS, jIHaDtaHvIS, latlh je, jIDoy'choH.
While I run, work, drive, study, etc, I become tired.

I know that the klingon {latlh je} is essentially "and others", since
the {latlh} is a noun (so one would expect it to being used only in
reference to nouns..), but I'm wondering whether the {latlh je} can be
considered a "set phrase" something like the {tu'lu'} which we never
write as {lutu'lu'}.

No data for an answer. I would expect it to only work with nouns, since it's got a je with it, but that's not any kind of proof.

-- 
SuStel
http://trimboli.name