I'm offering answers below in-case others can find value.
Maybe I'm going about this the wrong way and should start with basics...
1) Does MinerGate GUI 5.0.3 x64 (latest on site) work with Nvidia GTX 970?
Claude from MinerGate support confirmed support in GUI client.
2) If GTX 970 is supported through the MinerGate GUI, can someone please provide an idea of hashpower for a single card so I have a baseline to measure my system against?
3) What logs can I review to get more information about how things are working under the hood? I'm guessing that even though my menu's do nothing, the logs still exist and I should be able to find them?
The logs are available in %localappdata%\minergate\log
The log menu didn't work because no app was associated with the .log extension on my PC. This has been corrected.
4) Assuming everything is working correctly, should my hashrate from both CPU and GPU be combined in the website's dashboard?
I'm running an i7 with 6 physical cores, 12 logical cores on Windows 8.1 with 16gb ram. This system is water-cooled, and I have real-time monitoring of power usage and rarely tip 50% of the power supplies capacity. All drivers are current. Daily gaming at max settings, as well as running CCminer for over a year on this system without any issues.
I've tried running the MinerGate CLI, but have not spent enough time with it to know much more other than it appears to detect CUDA 5.2.
I've been sitting in the English chat tool on minergate.com as well if anyone wants to do some real-time assistance.
Out of curiosity, did you ever try a clean install after all or did you solve the problem?