First off, I'm not sure what happened in this KGS game, but I (white) did horribly. I resigned shortly after this, but, how should one respond to this invasion (Black at Q3) in later games? It seemed like a very arrogant play, and I wanted to punish it, I just had no idea how.

You can put links to KGS game archive links, this is easier to check

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Geek of all trades, master of none: the motto for my blog mostlymaths.net

Hi Paradox, even without looking at the SGF,
we can just look at your screenshot, and immediately
notice something is off on the lower left corner for W:
at least one stone there seems misplaced: the empty triangle
there is suspect.

After the game, have you studied the joseki starting
with Kitani's head bump ?

Killing the invading stone seems a little bit unrealistic but you can enclose black and make a big wall yourself.
Living often results in gote and as white you could continue with a stone around R12 to try make use of your wall...

The 3-3 looks about right, especially as the O-03 stone already has a 2 space extension. If Black runs, White will have a lot of fun attacking the weak group. (Black should cut his losses and play elsewhere.)

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The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

If you thought q3 was very arrogant, what do you think black thought of e18 and q1?

Instead of focusing too much on how awful you think your opponent's moves are, try to focus on making your moves the best you can.

Not enough, clearly. I just looked it up and see that I definitely didn't do it right. Joseki isn't something I'm very good at. I probably should have stuck with a simpler response.

Thanks for the advice though, everyone.
Especially Uberdude. Really sound advice. Especially because I didn't make this post specifically to try and make my own moves better.