I want to know what construction I should use in order to form sentences like "I saw you crying", "I caught him trying to steal smth" etc.
Literal translation for "i saw you crying" (participle) would be "я видел тебя плачущим",
while "я видел, как ты плакал" means "i saw, how you were crying" (verb) or "i saw, how you cried"
In my opinion, the second form is used more often because its simplier, but both have in most cases the same meaning.
Based on context and intonation, the second form "[...] как ты плакала" could also refer to the circumstances of how someone cried.
Better example: "я видел, как ты помыл посуду".
Translation: "i saw, how you washed the dishes", which means e.g. he washed it very bad. So the importat thing here is, how сomething happened.
Answer "Я видел тебя, когда ты плакал(а)." by Dmitry literally means "i saw when you were crying", which implies some contextual reference to the time.
Better example would be "i saw when you came home", so the time is important here and not the fact of coming home or seeing someone coming home.