Is it possible at all to use the LG UltraFine 4K display with my MacBook Pro (Retina, 13-inch, Mid 2014)?
No, this is not possible with LG's 4K Display since it is usb-c only and not thunderbolt.
Apple officially states that you are able to connect a LG UltraFine 5K Display since the thunderbolt 2 to 3 adapter works to connect them:
3840 x 2160 @ 60Hz
With the Apple Thunderbolt 3 (USB-C) to Thunderbolt 2 Adapter and a Thunderbolt cable, you can use the LG UltraFine 5K Display at 4K and lower resolutions with these Mac models:
Mac Pro (Late 2013) macOS 10.12.4 or later is recommended.
MacBook Pro (Retina, 15-inch, Mid 2014) and later
MacBook Pro (Retina, 13-inch, Mid 2014) and later
iMac (Retina, 27-inch, Late 2014) and later
iMac (Retina, 21.5-inch, Late 2015)
iMac (21.5-inch, Late 2015)
MacBook Air (13-inch, Early 2015)
MacBook Air (11-inch, Early 2015)
The necessary adapters for the 5K Display can be found here:
You can add a Thunderbolt 3 port to a Thunderbolt 2 computer by using an Apple Thunderbolt 3 to Thunderbolt 2 adapter and a Thunderbolt 3 device that has two Thunderbolt 3 ports (such as a dock, a raid, or PCIe expansion box). The second Thunderbolt 3 port can be used with a USB-C display. This means the LG UltraFine 4K display should be usable (but only if the computer supports 4096x2304 60Hz SST or 4096x2304 48Hz SST). The port can also support a Thunderbolt 3 display (limited to Thunderbolt 2 bandwidth), DisplayPort and HDMI adapters, and USB 3.1 gen 2 (10 Gbps) devices.
The Thunderbolt 3 port when originating from a Thunderbolt 2 computer can support one 4K 60Hz display or two 2560x1600 60Hz displays. The remainder of the 20 Gbps bandwidth can be used for PCIe data.
While Apple's support document HT207448 "Use the LG UltraFine 5K Display with your Mac" says that the MacBook Pro (Retina, 13/15-inch, Mid 2014) can support 4K (on the UltraFine 5K Display to be specific), the corresponding Technical Specifications documents for the same MacBook Pros makes no mention of 3840x2160 60Hz capability.
The Apple support document HT207447 "Use the LG UltraFine 4K Display with your Mac" only mentions USB-C and Thunderbolt 3 Macs.
The Apple support document HT206587 "Using 4K displays, 5K displays, and Ultra HD TVs with your Mac", does mention 3840x2160 60Hz support for the MacBook Pro (Retina, 15-inch, Mid 2014) (having NVIDIA GeForce GT 750M graphics) but not the 13-inch (which only has Intel Iris Pro Graphics).
Maybe (but unlikely) the UltraFine 5K Display is using an MST mode for it's 4K mode (as it does for it's 5K mode). Both the 13 and 15 inch 2014 Retina MacBook Pro definitely have the bandwidth for that. They support two 2560x1600 60Hz displays which is a similar amount of bandwidth required for a single 3840x2160 60Hz display.