Connecting a MacBook Air with Thunderbolt 2 to a Monitor with HDMI and USB-C

Solution 1:

I wouldn't recommend any of those. Instead get an ordinary mini-DisplayPort to DisplayPort cable and connect the monitor with a plain cable instead of adapters.

The 2015 MacBook Air cannot be charged over the USB-C port in any case, so you cannot benefit from that feature of the monitor right now. However in the future you might have a newer computer, and then the USB-C port with Power Delivery might come in handy.

Solution 2:

The LG 27UK850-W is not a Thunderbolt monitor. It expects a DisplayPort alt-mode signal from its USB-C port, so proposal #1 definitely won't work. While the Thunderbolt 2 port in the MacBook Air natively supports mini DP output, as soon as you connect it to Apple's Thunderbolt 3 (USB-C) to Thunderbolt 2 adapter, the signal coming out of the other end is purely Thunderbolt.

With the second proposal, it will work but is not ideal. USB-A display adapters typically rely on a tech known as DisplayLink, which is what enables USB to output a video signal. Not all adapters support your monitor's native resolution and refresh rate, so if you were to go this route, best to carefully check the adapter's maximum supported display mode. Some adapters top out at 3840x2160 @ 30 Hz, which is fine for watching movies, but will feel noticeably laggy with basically any other task.

As @jksoegaard has said, getting a regular Mini DisplayPort to DisplayPort cable will be the best option, cost and image quality considered. Is there any particular reason why you are considering your proposed approaches?