i5-7400 is a 7th-gen Kaby Lake CPU, and the H110 would need a later BIOS revision to support that CPU. It's difficult to determine if the board already has the later BIOS update. There is a chance that it does not, in which case you would need a 6th-gen CPU to install the BIOS update.
It would be best to get a B250 motherboard for the 7th-gen CPU, or a 6th-gen CPU for the H110 motherboard to ensure compatibility out of the box.
The i5-6500 is just as fast as the i5-7400.