Professor said no current flows to ground
In your example, the battery and resistors are floating as a separate circuit, except the tie to ground at one point. Because there is only that one ground tie, no loop to ground is formed, so no current flows into or out of ground. It’s the ‘loop’ part - or lack thereof - that your professor forgot to tell you about.
Anyway, the question is only interested in current. We also know we can ignore the ground tie (no current flows in it), it's a red herring. These two facts in mind, we can conclude that the only current that is flowing in the circuit is the loop formed by the battery and 2 resistors.
Kirchhoff's Current Law tells us that in a loop like that, the current is the same at all the points in the loop.
So the only sensible answer is (E), because all the currents are equal.
On the other hand, you raised the question about overhead power lines. These are referenced to ground. Because of this, if you are unlucky enough to touch a downed power line, and also are grounded, you complete a current path via ground back to the utility, and possibly, you die.
On the other, other hand, crows can land on power lines and survive just fine. Why? They practically mock you when they do that. That's just what crows do. Oh, you meant, why don’t they die? Crows, just sitting there on the wire, doing their murderous crow thing, don’t complete that path to ground that would turn them into smoldering corvid cinders: they only touch the wire, but are otherwise insulated by the surrounding air. No current flows through their insouciant bodies, even though they are at the wires’ high voltage potential.
As it so happens, humans can do this high-voltage party trick too, with a bit of help from a helicopter (so take that, crow!):

From this video: https://youtu.be/9YmFHAFYwmY (World Channel, https://www.youtube.com/channel/UCp7jpKjIOLFA1j3atWNJAKA)
The takeaway: no current flows to ground if there isn't a complete path that forms a loop back to the power source. Or, you're a crow.