That’s a good idea, to see if your local camera shop could take care of it for you for a reasonable price. That would eliminate a lot of the waiting and cost of shipping!
No, I wouldn’t use the cleaning kit I gave you to clean the sensor. You could give it a try, but I think that brush harbors dust and would add more to the sensor as you brushed it than it would remove. The sensor cleaning brushes are made to attract dust which is then blown out with the Rocket Blower (or canned air) before it’s used again. However, you could try a Q-tip with the lens-cleaning solution….some people do that, but I have never risked it. I don’t think it’s worth the risk using DIY tools.
And no, the dust inside your camera on the sensor wouldn’t have been caused by a car passing by on a dirt road….unless you happened to be changing lenses in that moment. The dirt you see has nothing to do with the lens. It’s all dirt on the sensor.