I seriously doubt that you can get any noise problem by grounding the coax. Except, of course, during an actual lightning strike and then, you probably will be worrying about bigger things.
I would use the shortest route possible to run a heavy ground cable from the bottom of the mast to a ground rod. And let the mast extend a foot or so ABOVE the antenna. As for the coax, I doubt that any cautions that you can take will either protect the electronic circuit it is connected to or provide any additional safety to your person than the grounding provided by the TV/receiver it is connected to. If you really want to, ground or attach a spark gap device at the point where it enters the house and ground that to the same ground rod via the shortest path possible.
Lightning protection depends on providing a VERY LOW RESISTANCE path for it to take. Running a wire to the other side of the house is counter productive. Keep it SHORT and use heavy gauge wire.
Lightning is a very strong electrical impulse and once it gets into the coax, there is likely no stopping it as far as protecting the circuitry is concerned. Best bet is to provide a much lower resistance path down the mast and the ground rod. It probably will not save the electronics where only a few tens of Volts will be a disaster, but you and your family will be well protected by that low resistance path and the additional, low resistance path of the house wiring's safety ground through the chassis.
As for noise, you will never see it. Digital signals either work, or fail completely. It is called the waterfall effect: good until it fails and then it very quickly fails completely. Just keep the connections tight and put "rubber" boots or some silicone sealer over the outside connections to prevent corrosion. Also use drip loops so the rain water does not run into the connectors.
Off Topic; I an installing a roof top tv antenna. Everything I read says to ground the antenna cable before it goes into the house. Problem is my ground rod is 50 feet away on the opposite side of the house. Seems like a poor idea to run a ground cable under the house for this safety feature.
Would it be okay to instal a separate ground rod at the antenna entrance. Or will this cause noise issues since the tv will be attached to a separate ground point. I suppose I good run a wire between the two ground rods to tie them together.
Very seldom any lightning here, never touches ground just cloud to cloud.
Bill D