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.
Grounding Electrode System
You're going to have to install a grounding electrode system. This is typically done by driving two ground rods spaced at least 6' apart, and connecting the rods together using a properly sized bonding jumper. Then you'll install a properly sized grounding electrode conductor, from one of the ground rods back to the main grounding bus in the main panel.
Bonding Water Piping
You may be able (required) to use the abandoned underground copper pipe as a grounding electrode, if it meets the criteria specified in the code. If the water is supplied to the building through metal pipe, you'll also have to bond the metal supply pipe to the grounding electrode system using a properly sized bonding jumper.
If the plumbing within the building will not be metallic, you don't have to bond it.
Bonding Gas Piping
As for bonding the gas pipe, according to the National Electrical Code, metal piping systems that are "likely to become energized" must be bonded (250.104(B)). So if you have a gas water heater that has no electrical connection, then you don't have to bond the gas piping. If you have a gas fired furnace, then it's possible that you'll have to bond the gas piping.
However, the bonding jumper only has to be sized to the rating of the circuit that is likely to energize the piping. NEC also allows you to bond the piping to the equipment grounding conductor, of the circuit that is likely to energize it. Which means the gas pipe feeding the furnace, can be bonded to the equipment grounding conductor feeding the furnace. And if the gas piping is isolated from the furnace electrical (not likely to become energized), you don't have to bond the piping at all.
In the case of the gas/electric range, again you can use the equipment grounding conductor of the circuit feeding the appliance, to bond the gas piping (if required).
Best Answer
The pipe only needs to be bonded at 1 place, usually this is at the meter. If it is bonded there the entire system is attached to a grounding bond and would be legal.
The gas pipe is bonded to prevent it from becoming electrified by a short. It cannot be used as a grounding electrode but the grounding electrode system needs to be attached someplace after the gas meter. I would check there as that is where the bond is usually located.
If you have lead pipes I would want to drive supplemental ground rods and get rid of the lead pipes (I hope you don’t drink the water). The reason to have a pair of additional driven grounding electrodes is a good idea for what happened to a tract of homes in my area. All the old pipe was removed and new non metallic pipe installed. A customer called because getting shocked. I discovered she had no ground and the pipe was plastic up to the house. Their utility did this to every home and the only grounding electrode was the water pipe. After installing a pair of rods she quit getting shocked. If that same case happened to you your gas pipe would end up being your electrode not safe and a code violation.