I'm unable to hear sounds on right side of headphone jack
 

I'm running Arch + Gnome.

Using Speaker testing in GNOME, I was able to find out the following:

1. Left laptop speaker and right laptop speaker work.
2. Left earphone works. Right earphone does not.
3. The earphones work properly on other devices (phone).

How do I know if this is a hardware issue with my jack, or if there's some software problem? Thank you!

It is very likely to be the jack. Make sure to plug them in all the way. Also check the volumes and make sure the volume one one side is not turned all the way down.

I've done both of these. Does this mean that I can never use this jack for audio again?

Yeah it could be the female part of the jack that is broken, I mean if the male part works on other systems.

Boot to a Live CD of something (Mint would be a good choice). If the headphone port doesn't work properly under that, it's almost certainly a hardware problem.

Is this laptop or a desktop? If it's a desktop, replacing the port should be a simple matter.

Open up alsamixer in a terminal, and find the slider for the headphones. Look at the bottom of the bar where there are two letters side-by-side. If it looks like "OM" then use the arrow keys to select that slider and press the period key. It should then change to "OO".

If that is not the case, then it's likely a broken headphone jack on the laptop. If you're brave and wouldn't mind seriously messing up what's left of the headphone jack you could take it apart and see if there's any obvious broken connections (especially the solder points between the headphone jack and the motherboard) you could try to fix it yourself.

Hope this helps!

The suggestion of using alsamixer helped! I don't know why, but one of the channels was nearly zero. I have no idea what caused this since I didn't see the setting anywhere in Gnome (the balance was proper), and I was able to fix it only after installing alsa-utils (for alsamixer).

I can see now that it wasn't this, but using canned air to clean out the headphone jack very often resolves problems like this.