MacBook Pro Display – Is It Possible to Wirelessly Connect MacBook Pro to LG Monitor?

It is not possible. You can only use an iMac as a target display.

Yes, Thunderbolt 2 is indeed compatible with DisplayPort 1.2. On top of that you should have no issue running a resolution of 3840x1600 on an external monitor via your MacBook. You will likely have to set the resolution manually though. After connecting the monitor, just head to System Preferences>Display>Your Monitor and hold option while clicking the "Scaled" radio-button to show a full list of available resolutions.

However do be wary that Night Shift is not officially compatible with any displays besides the internal Mac displays, the Apple Thunderbolt Monitor, and the LG UltraFine 4K/5K monitors. There is a common problem on unsupported external monitors where Night Shift will become permanently enabled, and will even stop Night Shift's scheduling and intensity settings from working properly on the Mac's internal monitor. I wrote a guide about fixing this by tricking your Mac into thinking your external monitor is internal. This will also prevent another problem where some external monitors get stuck in mirroring mode on newer versions of macOS. You can find that here: Fixing Night Shift incompatibility with external monitors