Don't forget, brother that the Holy Spirit came in literal form in more than just the dove. He came in literal tongues of fire in Acts. The dove represents just one aspect of His character.
Also, I think we are affected too much by the media. Yes, there are a few species of doves that are white like the ones we always see on television being flown at weddings and in movies about Noah's Ark.
But doves come in a WIDE range of colors. The mourning doves and mountain doves referred so many times in the Old Testament and which have MUCH symbolism weren't white at all. They were brown and gray and blended in with their surroundings.
And just google "fruit doves" to see the hilariously crazy colorations God painted on the doves of jungle areas.
I've done quite a bit of reading this morning via biblegateway on biblical references to doves. I think I've read and studied them all, but I might have missed one or two. Just a couple of things.
The dove references of Song of Solomon 1, 2, 4, 5, and 6 are pet names give in romantic conversations and represented the purity of love as God intended and in two of those references symbolize the physical beauty of a woman.
The "wings" of doves references in Psalm 55 and 68 represent salvation - both literal and figurative. An escape, if you will, from this world's tribulation and stains of sin.
Isaiah 38, 59, and Ezekiel 7 refer to mourning doves and their grief. These passage symbolize grief from sin, loss, captivity, and more.
Hosea 7 and 11 equate a dove with mankind's helplessness or "silliness" as it says in the Kings James. The half-tribe of Ephraim is described as a silly dove flitting around helplessly with no purpose and ultimately being captured and eventually "flying back" in the restoration.
The future church is prophesied about in Isaiah 60 as being "doves flying as clouds to their window". This, in context, speaks of the innocence and enormity of the future church.
Doves come in many colors and represent many things in the Bible - both literal and figurative. The Hebrew word for pigeon is the same as the Hebrew word for dove, also.
I just don't see anything prophetic or "encoded" in the Noah's ark account of the dove. We don't even know what color that dove was and it was merely just to test for vegetative growth.