How Do I Bond My Antenna to Home Electrical Ground?
The purpose of an earthen ground is not to direct lightning. It's unlikely that you'll have lightning running down a wire routed through your walls, unless you have other more severe electrical problems in your residence. Your home's electrical wiring should have a current earthen ground "somewhere" and all devices using that electricity are conductive to that ground.

Having an independent earthen ground in the manner you describe ensures that your additional device, in this case, an antenna has the electrical potential set to match the rest of the system. If you did not have an earthen ground for the antenna and received outside electrical impulse (lightning), the charge would then attempt to travel through the signal cable into the electronics attached to the antenna and from there, through your house's electrical ground to the earthen ground. It's unlikely it would travel that far, however, as the electronics would take the brunt of the force and go up in smoke.

An independent earthen ground provides for a safer path for the antenna, the mast of which should be bonded to the ground. You can also bond the coax from the antenna to that ground with a grounding block. If you have satellite coax without a grounding block, install one and bond it to the same ground. 

There is no reason to run the ground wire into the house, unless you have devices within that are not grounded to the house ground system.

The primary objective is to have every device at the same electrical-potential-level of the ground rod. The outer conductor of the coax will provide suitable ground connection to the electronics within, for both the satellite equipment and the OTA equipment.

If you were to take a severe enough strike on your antenna mast that the earthen ground could not safely dissipate it, and the "excess" traveled on the coax, it would likely melt the coax and prevent travel into the house. As a former cable television technician in Florida, I found many instances where the house ground was not bonded to the cable tv block ground. The lightning strike/surge that traveled into the residence on the power line exited on the cable tv ground, burning up the electronics in the television. Back in the 80s, it wasn't practice to bond the house and cable tv grounds together. That has since changed and destroyed television are rarer now.

Regarding the ground wire running in the dirt, it's not going to change anything with respect to electrical potential. I don't know NEC codes, but I'd be surprised if there was anything of concern there. If you could drive #6 copper straight down 8 feet, you'd have suitable grounding. It's easier to hammer in a 1/2" rod though.