Electronic – What does the arrow going through the battery symbol represent

The first two calculations look good. The first one is a little ambiguous on the problem's behalf because the current is more than 3A, but not a short circuit so we don't explicitly know the series resistance of the battery. I also agree with your answer for the last one but let's understand why.

Let's do the calculation exactly the same though for c. We now have a zero ohm resistor. And since we know a short circuit condition is present we know the equivalent series resistor will be 2Ω.

$$ I = \frac{12V}{\left( \frac{1}{\frac{1}{10} + \frac{1}{10} + \frac{1}{0}}\Omega + 2.0\Omega \right)} $$ Simplifies to: $$ I = \frac{12V}{\left( \frac{1}{\infty}\Omega + 2.0\Omega \right)} $$ Finally we can see the the equation does simplify to $$ I = \frac{12V}{2.0\Omega} = 6A $$

Appending to answer your last question which I missed. Even if you don't explicitly know what the purpose of the infinity resistor is doing in the circuit solve it the same way: $$ I = \frac{12V}{\left( \frac{1}{\frac{1}{10} + \frac{1}{10} + \frac{1}{\infty}}\Omega + 2.0\Omega \right)} = \frac{12V}{\left( \frac{1}{\frac{1}{10} + \frac{1}{10} + 0}\Omega + 2.0\Omega \right)} $$ Which is what you had.

**Side note, current is usually denoted with an 'I' not an 'A'.

This looks like a spark gap.
(You could also look up gas discharge tube (GDT), which is similar to spark gap.)

The likely purpose of this component in your circuit is to protect the rest of the circuit from lightning strike and/or ESD. Spark gaps are often the first line of defense against high voltage.

Advantages of spark gaps: Once the arc is established, a spark gap acts as a crowbar, and it can dissipate a lot of energy. Because of that, spark gaps are used for protection against high voltage high energy threats (lightning, defibrillator). The parasitic capacitance of a spark gap is low, so it doesn't affect the signal. A spark gap can be formed as a PCB feature for free; it doesn't require an additional component in a BOM.

Weaknesses of spark gaps: They fire at a high voltage, hundreds of volts, and the firing voltage is not well reproducible or predictable. To address this weakness, there's usually another overvoltage protection device (such as TVS) in parallel with a spark gap. This additional device clamps at a lower voltage.

related:
What is this component and what is its use?
https://electronics.stackexchange.com/a/28959/7036
EEVblog #678 - What is a PCB Spark Gap?