If you don't know what the power supply is, there is no way I could know if you'll need to upgrade. Nvidia recommends a 500W PSU, but you also need a quality PSU.
Check this out: http://www.tomshardware.com/reviews/best-psus,4229.html
I know that Photoshop can't take advantage of two cards. I think you really can only use the 970 as a dedicated PhysX processor in games. Otherwise, it will just be taking up power and space.
But yes, you can use the 1070 in your current setup.
A curved screen at such a small size will look pretty terrible from far away, so I would not recommend going with a curved monitor.
An expensive alternative is to get the monitor, but also buy a TV to hook up to your computer to watch movies.
You cannot have both games running in true fullscreen. But you can run them both in borderless windowed mode, which is basically fullscreen without giving the program exclusivity.
Have you tried using the bootable USB? Because if it does, that might rule out a hardware problem.
If you have another hard drive laying around, or another large USB drive, you can try installing Windows on that, too.
Does it have an AUX input? If so, does sound come out of the speakers when using it?
Also, does the HDMI audio work if you plug it into another display, like a TV?