I would start with El Capitan or Sierra first. Don't think it's a good idea to try Mojave or newer on a 775 Core 2 Quad system. Some have done it but it gets more complicated. The Core 2 Duos/Quads don't support SSE 4.2 instruction set. A few Core 2 Duos from 2009 (Wolfdale) would work with High Sierra but those only had SSE 4.1. 2009 C2Duo iMacs still had support in High Sierra.
Your Intel graphics won't work, only HD3000/4000/4600 iGPU will work with El Cap, Sierra or even High Sierra and your CPU doesn't have an iGPU. You can use a low cost supported Nvidia Kepler card. What OS version you choose will make a difference. Some older AMD cards will work as well but they tend to cost more.
Preamble for those new to Nvidia graphics card choices for a Hackintosh 1. You will only have success choosing macOS compatible graphics if you remember that: macOS is not Windows 2. Apple incorporates all their graphics drivers into the Mac Operating System. There are only drivers in macOS...
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