I think that the manual shows that the LHT65 does not support an external antenna. I don’t know if you can extend the wiring for the external temperature sensor, but the humidity sensor is built-in anyhow.
But like you noticed: the gateway seems to be close by, and you could always re-use that device for something else.
Also, I wonder if it needs any programming/configuration. If yes, then you’ll also need a USB-to-TTL adapter (quite cheap), to connect the data cable to a computer. I think by default it uses OTAA to join the network (TTN), so you should be good to go without any configuration. However, you might want to read about it possibly not having its own unique value for AppEUI, for which you might need to reconfigure it using that cable. The data cable might also be useful to see debug information, assuming it outputs that.
With the proper decoder in the application’s Payload Formats in TTN Console, yes. (Minor detail: the decoder’s usage of .toFixed(2) will result in text, not in a number. Easily fixed.) But according to the manual it also seems to be supported by Cayenne, so that’s even better.