For all of these examples except the fourth, I would choose victory. The reasons why are a little different for each, so I'll do my best to explain them individually. The differences between these words are few and relatively minor, but one to take note of is that a triumph is very explicitly made against something or someone in particular, whereas a victory can be achieved in a much more broad sense, often with no specific, discernible enemy. When in doubt, remember that a victory can be shared by all parties. A triumph could too, but usually that's not how it's used.
1) The signing of this contract was a great victory for our company.
For both the above and below examples, as well as most political/economic settings in general, it can be seen as rather inappropriate to use triumph. In many contexts, a triumph requires a very specific winner and loser -- someone comes out on top, and the other will likely suffer for it. In business, this outlook would be rather frowned upon, especially given the context: two companies sign a contract together, mutually agreeing upon a set of terms and forming a legal bond to uphold them. If either one "triumphed" over the other, then something has likely gone wrong.
2) That agreement was a tremendous diplomatic victory for France.
Similar to the above question, though triumph probably has a little more leeway. If the other party here is another country, this may be very inappropriate, but if they are, say, a terrorist organization, it could be a very fitting word to use. Without more context, though, victory makes for a better blanket term.
3) The team returned home in victory.
Personally, I think that with some added information this could much better suit the use of triumph, but at the moment it is too vague. As stated earlier, a triumph often requires a specific other party, whereas none are specified here. Had the sentence read "in victory against their long-standing rivals," then I'd switch my vote to triumph.
4) Do you remember Spain’s 3–2 triumph over Russia in last night’s game?
Similar to the above, now that there is a very clear distinction between opponents, there is no harm is calling the win a triumph. In most sports, the entire purpose is to beat one another, for a clear winner to be designated -- so this word works rather well here.