This construction is probably easier for Spanish speakers to understand. Spanish and Catalan are the only Romance languages (as far as I know) that have kept the verb "solere" meaning "doing something on a regular basis". The Latin sentence would be "Las comadrejas no suelen cocinar" in Spanish.
EDIT: Also Italian has the verb "solere". You live and learn.
German uses the past participle of wohnen (to live/reside probably some connection to the noun wont) - gewohnt like English does with use. I don't think the logic is super foreign or special. At least I think I don't miss some magic of this solent/suelen.
Wiesel kochen gewöhnlich nicht. Weasel do not usually cook.
Wiesel sind nicht gewohnt zu kochen. Weasel are not used to cook.
I guess the meaning is more accurate transferred in the second version but I can't describe to myself why it feels somewhat different.
Why is 'solent' placed at the end of the sentence?
The conjugated verb typically goes to the end of the sentence in Latin.
Usually is an adverb in English.
I'm not quite sure what you mean here. "Solent" is a verb in Latin; in particular it is an auxiliary verb like "can" or "should" in English and so usually goes with a second verb ("can go"; "should eat"). We don't have a verb in English that has the same meaning, so we have to approximate it with other words; roughly "soleo" means "to be used to [doing something]" or "usually [do something]."
There is a verb to use. eg I have used all the butter.
"Soleo" means to "be used to [doing something]," not to use something.