Facts that could save your life

I did tae kwon do for a while

I'm starting Brazilian jiu-jitsu

Aren't you fancy

Always

Or, or, gouge out their eyes with your thumbs

One reason why I'm glad I have very strong legs and wear steel toe work boots, I can deliver one hell of a ball busting kick or any kind of kick for that matter. Improvise folks make the world your weapon!

I guess I have really primal ideas about self defense

Yeah
Gouging out eyes is old-fashioned

My other thought was to bite them, specifically on the neck.

Kinky

It's really fucking hard to gouge the eyes. Like...almost impossibly hard in a rock n' sock fight. Gouging only really happens when someone's in a submission and can't defend against it. And it's also disgusting to perform.
Ball kicking is also surprisingly hard to do. It's a relatively small target with several quick and easy ways to defend against. In order to land a crippling ball strike, you either have to completely break through someone's guard or land an unguarded strike going in a certain direction (I forget how); it's either you need to crush or slap, I don't remember. But anyways, it's really easy to instinctively defend against this.
What you should do is go for the kidney/gut and jaw. The gut is a rather large area to hit and impossible to perfectly guard the entire area at once so a good hit will slip in and wind them. A jaw strike dazes.
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Remember children, ninjitsu is bullshitsu. You wanna be a ninja? Go join the SEALS or Rangers or some other special force.

The absolute law of combat is weight and strength win. Skill is simply an application of strength; if your strength is overpowered it doesn't matter how you apply it. If you're outweighed, run the fuck away.
Also, fight simply. Nothing fancy. You're not trained to do any bullshitsu so your body will not act exactly the way you need it to. Defend and run.

Dont forget SING
Solar plexus
Instep
Nose
Groin
Four of the most sensitive areas

And four very small targets. Like I said, it's all very well and good to try and aim for "critical" strikes but don't make them a priority. Just aim around them and you'll hit them eventually; the four locations you listed are all in and around areas that a good strike can still do damage even if they miss the sweet spot. Solar plexus is near the intestines and kidneys, good for winding. Instep is on the foot, near the ankle. A kick hurts a fuck ton, maybe not crippling but still very painful. Nose is near jaw and on the face. Groin is on pelvis and near gut. There's a reason why leg kicks aren't allowed in Tae Kwon Do, it's really dangerous.

And I don't necessarily mean to denigrate self defense mnemonics. I just want people to realize that street fighting is a purely chaotic environment and any plan you have as an untrained person is very likely to not work, even with a compliant partner simply because of your own physical limitations (flexibility and strength mainly). We can't do bunshin no jutsu, throw shruiken around the back of a tree, or see chakra points; fighting isn't magic.

What about biting?

Very good. Painful, leaves a trackable mark, easy to do if they try to work your head. But of course, don't try and bite their shin or anything. Use common sense.

Good

Human bites are prone to infection
So, if you want to do damage, bite them

IIRC cat wounds are even worse in that they're unusually septic.