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Again, not a physicist, so here's a bunch of words that sum up to "maybe."
With the object moving so fast I'm not sure we'd notice anything much. We would only be in it's gravitational field for a very short time, but it might be long enough to change Earth's orbit, someone with better math skills will need to field that one.
As for heating the Earth, again that's a maybe. Gravity is stronger the closer you are to the center of mass. So the near side of the Earth will feel the pull of the object much more strongly than the further side. That will make the Earth want to stretch towards the object as the near side falls towards the object faster than the far side. It would be very slight, think egg-shaped but not to a noticeable degree, but it could be bad enough.
This is called a tidal effect and would generate some heat if we're in the gravity well of the object for long enough. It would also cause fault lines to pop all over the globe. The object would shoot by very quickly though at 99.9999c so we might be spared the worst of the effects. Again, someone with better math skills could give a more accurate answer.
FYI tidal effects are why the moons of the gas giants aren't frozen ice balls. The constant flexing as they orbit their planet generates tremendous amounts of heat.