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.