Learn Russian: Мы надеемся одержать победу в суде.

Yes, Мы надеемся победить в суде means exactly the same thing. The phrase одержать победу (to achieve victory) is a formal analytical predicate. Using a light verb and noun combination like this makes the sentence sound more formal, official, or dramatic, which is very common in legal, journalistic, or political contexts.

What does the verb одержать actually mean, and can I use it to mean 'achieve' or 'win' with other words?

Outside of a few set phrases, одержать isn't really used on its own in modern Russian. It historically meant 'to hold back' or 'to gain the upper hand.' Today, you will almost exclusively see it in the fixed collocation одержать победу (to achieve victory) and occasionally одержать верх (to gain the upper hand). You cannot use it as a general translation for 'achieve'.

I sometimes see на суде. Why is it в суде here?

Both are grammatically correct but have slightly different nuances. В суде generally refers to the court as an institution or a physical place (like winning a legal battle in the court system). На суде refers specifically to the event or the process of the trial itself (for example, testifying at the trial).

Why are we pairing the imperfective verb надеемся with the perfective verb одержать?