Will this mix of Shimano 10 speed components work together? My goal is to cobble together a 1x10 groupset using some Shimano 105 5700 shifters. 

From what I have read, I think that these components will work together, but I want to be sure before purchasing: 


Shimano 105 5700 10 speed shifters
Shimano GRX RD-RX400 10 speed rear derailleur 
Shimano Deore HG50 10 speed cassette 11-36
Shimano Deore T610 10 speed rear hub


I will be in a 1x configuration, so no front mech is required. 

Will this mix of Shimano 10 speed components work together?
Sorry no, the 105 shifters and GRX derailleur will not work together.

Older Shimano 10 speed groupsets use the 'classic' 1:1.7 actuation ratio (length cable pulled / lateral cage movement). 11 speed groups required a different ratio to work properly. The newer 10 speed groups - Tiagra 4700 and GRX 400 series - both use the same ratio as the 11 speed groups, and are therefore not compatible with the older 10 speed groups.

Shimano publishes extensive specifications an compatibility information for their products. The latest compatibility information is here: https://productinfo.shimano.com/#/com. If you look at road rear drivetrain compatibility you see that the GRX RX400 derailleur only works with Tiagra 47xx, 'Non series' RS405 and GRX RX400 shifters.

You can see that the HG50 cassette is explicitly compatible with the RX400 derailleur.