first I have to finish digitizing the family slide collection with a scanner that isn't Windows 8 compatible.
Chances are the scanner might be Linux compatible - I inherited my dad's old printer when he upgraded Windows and no driver was available, it works fine with my Linux system. So you could maybe try a Linux live boot system (CD, DVD, USB) that won't modify anything on your hard drive, and see if the scanner works. Check the package list of the live system before downloading, needs GIMP and GIMP plugin for SANE (which is the Linux scanner system - you don't strictly need GIMP, but command line scanning is more awkward).
So, how many CUDA cores and how much dedicated RAM do I need for a decent GPU rendering experience? Would something like the NVIDIA GeForce GTX 745 4GB DDR3 (384 CUDA cores) do the job? Do I even need a dedicated graphics card? My brother-in-law was trying to tell me that integrated graphics would work -- is that true?
Integrated graphics would work fine for displaying the results of CPU rendering, and are increasingly more powerful, but for GPU rendering you really would want something much more powerful I think. Also, you probably want to consider double precision floating point performance on the GPU, something not typically covered in game-oriented benchmarks (single precision floating point is pretty much the standard there). You might need to look for science/engineering-oriented benchmarks. Double precision offers much greater zoom depth than single precision floating point, which could be important depending on what you want to render.
Fairly standard point: it's probably not worth going for top-of-the-range, look for best price/performance ratio (unless you absolutely need maximum performance and are willing to pay over the odds for it).
One more question -- will a modern graphics card support a seven-year-old monitor (with DVI-D and D-sub connections), or am I going to need a new one?
It probably is possible to get a graphics card with DVI outputs still, though HDMI is increasingly the preferred option. D-sub (analogue VGA if I understand correctly?) is obsolete by now I think. Otherwise the motherboard will probably have some integrated graphics too, so you could render on your fancy GPU and display the results on the integrated output - might be a nightmare to set it up though. Check the output connector specs before you buy. Another thing to look into is multiple outputs on your graphics card, in case you eventually decide that 1 screen is not enough.
EDIT: found some fp64 double precision benchmarks (seems AMD is doing better these days, so unless you absolutely need CUDA then maybe broaden your search from NVIDIA alone)
http://www.tomshardware.com/reviews/geforce-gtx-780-performance-review,3516-28.htmlhttp://www.theinquirer.net/inquirer/review/2162193/nvidias-gtx680-thrashed-amds-mid-range-radeon-hd-7870-gpu-compute