Does a ThunderBolt cable increase resolution on a monitor or just frame rate?
A) A cable cannot "increase" the resolution of one source to a destination. However, if the cable provided more bandwidth than your previous cable, the system would seem to upgrade the speed because of the extra bandwidth available.

Think of it in terms of a highway with cars. If your old cable is only a two-lane highway, you can only put a certain amount of cars on it. Now if your new cable is an 8-lane expressway, then you can put so many more cars on it and even though the speed of the cars does not necessarily increase, there are more cars and more bandwidth and less traffic.

B) As for the LG Electronics 34UM88-P, it supports thunderbolt 2.0 as well as DisplayPort - both of these are supported on the Mac Mini.

Source: https://www.lg.com/us/monitors/lg-34UM88-P-ultrawide-monitor