Basically yes, you can use a HDMI<->DVI cable to connect your PS4 to your monitor. You just need to make sure it's DVI-D (not DVI-I). The problem that comes with this, is that there will be no sound via the cable anymore, but you can still use your PS4 headphone if it is plugged into your controller.
As for the resolution, i think it depends on the game. Generally, the PS4 will still output its native resoluton which in most cases is 1920x1080 or 1280x720. If your monitor has a 16:9-mode, use it. If not, games might do one of the following: render in 720p and upscale as good as possible; generate black bars; cut off sides.
You can use multiple cords leading to one port with a DVI-switcher (I have no experience with these but they seem to work fine). Personally I think simply switching the cords manually to the device you want to use is less of a hassle than buying and setting up the switcher and cords that are needed but that's your own choice obviously.