When I'm writing to a person who didn't reply to a lot of emails I sent him, is it wrong to say "I sent you many emails this week but I haven't received any replies"? Must I say "I've sent you many emails this week, but I haven't received any replies"? It seems like the "sending" was completed, so I'm not sure if it's totally wrong to say "I sen you many emails this week".
Grammatically they're both correct. I think it's only the word many that makes 'I sent you many emails ...' feel awkward. Try saying the sentence with a number (eg 50) instead of 'many' and it feels perfectly natural.
It might also feel odd because send doesn't conjugate like normal verbs. It doesn't adopt the 'ed' ending and there is no difference between the 'I did' and the 'I have done' versions, unlike, say 'I saw' and 'I have seen'. It might feel better perhaps if send conjugated to 'I sent' and 'I have sended'. (The latter of which is of course, entirely incorrect.)
The past tense is used to describe an event completed at some particular time in the past. If that’s what you're referring to, then I sent is fine. However, you’re complaining about sending so many emails and still getting nothing in return. This is where the present perfect construction comes in. It is typically used to describe a past event that has relevance at the time of speaking. If that’s the case, as it almost certainly is, then I’ve sent is what you want. That, at least, is from the perspective of a speaker of British English. American English tends to use the past tense more than the present perfect construction, and a native speaker of American English may want to comment on this point.
That was the question you wanted answered, but there are a couple more points worth mentioning. In the informal context of emails it would be more usual to a lot of emails or lots of emails rather than many emails. The other point arises from the comment about the need for any to be used with a singular noun. This is not necessarily so. Any replies would be normal and grammatical if you were expecting more than one reply.