No, "остановиться" does not refer to stopping a vehicle specifically.
"Остановиться у родственников" = To stay with relatives (e.g., during your vacation)
"Часы остановились" = The clock has stopped.
What you hear from Russian-speaking mothers is one of the imperative mood verbs used to deliver the "stop it!" idea.
Перестань! Прекрати! = Stop it! Quit it!
Успокойся! Угомонись! = Calm down!
Хватит! Довольно! = Enough!
Хорош! Всё! (colloquial) = Enough! That's it!
The root of the verb "остановиться" is "стан-", which has the meaning of "being stationary".
"Пристань" = a pier (a place for ships to be stationary)
"Стан" = a camp (one of the meanings for this Russian word; a place where people stay)
The "о-" prefix in this case delivers the meaning of "becoming, making".
"Остановиться" = to stop (to become stationary)
"Остановка" = a [bus|train|taxi] stop (a place where bus|train|taxi become stationary)
It's a prefix, not a suffix. Here's a link that you might find helpful. https://en.wiktionary.org/wiki/Category:Russian_prefixes
Stay = "остаться/оставаться"
Can you stay for the night? = Ты можешь остаться на ночь?
Just like in English "Can you stop?" and "Can you stay?" mean quite different things, so do "Вы можете остановиться?" and "Вы можете остаться?" in Russian.
However, if you want to say "stay at a hotel" or "stay at my friend's house", it will be translated into Russian as "остановиться в гостинице" and "остановиться у друга/подруги". That's one of those language quirks, sorry.
Correct, you do need a qualifier here - "Do you want to stay with us | for a night | for a while | forever (if you REALLY like them)?" Without the qualifier, "Вы можете остановиться?" means "Can you stop?" On the other hand, "Вы можете остаться?" means "Can you stay?" and doesn't require additional qualifiers to deliver the meaning