Also, this may be a dumb question, but if I leave the 5770 in (for the occasional boot screen or OS problem), will there be any conflicts with having both it and an Nvidia card installed at the same time, particularly in Windows? (Specifically Windows 10)
Unless you have an extra power supply (or some other unconventional way to give power to the card), you cannot power both of those cards through the logic board. The Mac Pro's logic board only has two power connectors, and the Radeon needs 1 while the GTX needs 2. Your best bet IMO is to sell the Radeon 5770 (they sell well on eBay) and pick up an Apple card that does not need power connectors - a Geforce GT120 or a Radeon 2600XT. See this thread for more https://forums.macrumors.com/threads/dual-gpu-mac-pro-2008-how-to-power.1954282/