Apparently this CPU is a bad combination with my GPU; I have been told that they will not work together.
That CPU and GPU will work together just fine.
GPU's are designed to be compatible with most CPU architectures, including offerings from Intel and AMD. Ryzen is AMD's new platform, and just like any new release there were some kinks to work out in regard to memory management and driver / system compatibility. Ryzen CPU's have been out for a while, so as far as I know those issues are all resolved. I have several friends and co-workers running Ryzen systems, running a mix of GPU's from both Nvidia and AMD. No major or irregular problems reported.
No overclocking so do I need CPU cooler separate?
You do not need an aftermarket cooler for any Ryzen CPU if you're not overclocking.
Even if you are overclocking, the "Spire" cooler series (which come stock with Ryzen chips) are very good air coolers. I'd recommend you save your cash and use the stock cooler. You can always buy a big fancy one later if you end up wanting to do extreme overclocking down the line.
There will no intense programming or video processing; I will use the PC for some games here and there, but it will only be doing regular "home" tasks everyday.
So I know this part wasn't really a question, but I do have a recommendation or two.
First, the CPU you're getting is probably overkill for your goals. It offers some future proofing and flexibility, but as @SiXandSeven8ths mentioned already in the comments you'd probably get equivalent use from a Ryzen 5 series unless you're going to be doing VR gaming.
Second, if you do go with the Ryzen 7 1700, get the Ryzen 7 1700x instead. They're the same price, but the 1700x is a better "binned" chip. This means it'll be faster and more stable on average.
www.pcpartpicker.com is a good ressource to use. I've got your build started for you at the following link:
Good luck with the build!
Regarding questions from OP's edits:
I like the board in your PCPartPicker list a lot. I've had good experiences with ASUS motherboards and from what I've heard they handled the release of DDR4 and the new socket types with minimal speed bumps.
The CRYORIG C7 cooler will work just fine. It's designed to fit the following sockets according to PCPartPicker:
AM2, AM2+, AM3, AM3+, AM4, FM1, FM2, FM2+, LGA1150, LGA1151, LGA1155, LGA1156
I don't believe it has any LED's, or at least I can't find any pictures or documentation suggesting that it does. Most coolers that do feature LED lighting have them installed directly on the fans, which are replaceable anyhow. I'd say that you've got a green light to go ahead and get the C7, as the worst case scenario is replacing an LED enabled fan with a non-LED fan.
Please let me know if you have additional questions, I'll be happy to follow up. Best of luck with the build!