Is it wrong to say that the Holy Spirit proceeds from the Father and is sent by the Son? If so, why?
The first part of your statement "The Holy Spirit proceeds from the Father" is correct. The question that remains: is He sent by the Son only? Consider these two verses "But when the Helper comes, whom I shall send to you from the Father, the Spirit of truth who proceeds from the Father, He will testify of Me" (John 15:26) and "But the Helper, the Holy Spirit, whom the Father will send in My name, He will teach you all things, and bring to your remembrance all things that I said to you" (John 14:26). In one verse our Lord Jesus Christ shall send the Holy Spirit and in the other it is the Father who will send the Holy Spirit. So "sending" is not restricted to the Son.