The .CYSPPSTART command is used to manually start CYSPP operations in the configured mode (peripheral or central), but this is only necessary if you have changed the "enabled" setting with .CYSPPSP to something other than the default "autostart" value (0x02). Separately, the .CYSPPCHECK command only applies if you are operating in client mode and have already connected to a remote peer device, and you want to manually run a CYSPP-specific service/descriptor discovery to verify whether CYSPP-related GATT structure is present. Those commands are usually not required, especially not if you are still using the default configuration.
The 0x0107 result from the .CYSPPSTART command in your case indicates an invalid state, which is expected if you are already advertising (the default behavior on start-up). The 0x0502 result from .CYSPPCHECK is likewise expected based on the operational state of the module at that time, particularly that you are advertising as a connectable peripheral device, and not connected as a central device.
I would recommend reading through Section 3.2 in the EZ-Serial User Guide for explanation and usage examples for CYSPP.