What is a carbon steel pan?

Technically, carbon steel is an iron alloy that--contrary to its name--actually has a very low carbon content. This makes it relatively easy to form, infinitely more ductile, and slightly more conductive than cast iron. But practically speaking, you can expect a carbon steel pan to be appreciably lighter than cast iron, with better non-stick properties. In short: all the benefits of cast iron, but lighter and prettier.

How should I care for my pan?

If you're familiar with cast iron pans, these would benefit from a similar regime. I  am pretty laissez faire with mine. I like to wipe them out warm (if possible) and hang them up. Otherwise, just wash them without soap and dry them well. If they're to be used infrequently, I'd suggest giving them a light wipe of oil before putting away. The main thing to remember is that they're pretty much indestructible  and will last several lifetimes, surviving global calamity along with the jellyfish. Whatever you do to them, it can probably be undone.  

Why aren't they round?

In my architectural work, designs often require that I form a piece of sheet metal  and then trim all that wasn't part of the intended design-- much in the way you trim excess pie crust after laying rolled-out dough in a pie pan. Quite often, I found the remnants from these pressings far more intriguing than the formed shape itself, since the story of the deformation was most visible through what was cut off. In other words, the beauty and uniqueness of the process is often most evident in what it leaves out. The pans I sell now demonstrate the characteristic "swoop" I noticed after press-forming a bunch of octagonal sheet steel for some lighting fixtures a few years back. 

All that being said, the bases of the pans are actually round, so you've got nothing to fear regarding your quiches and your crepes.

Does the shape change the way they work?

 Absolutely  not. The only difference I've noticed is being able to scrape the back of a wooden spoon more effectively. Oh, and they tend to impress  people solely for how different they look from most other pans. So I guess they work better at dinner parties... but there's no hard science I can reference to that end (yet).

I've heard that carbon steel is nice but requires hours of scraping and a xylene bath before being able to cook with it, to say nothing of how to begin the seasoning process. Is this true?

Some of the large scale industrial producers of steel cookware do coat their wares with a variety of (nearly) inedible coatings. Being a small operation, I don't. All of my products come pre-seasoned with a thin coating of virgin coconut oil. This seasoning will build up over time, turning to a black iridescent sheen. You can cook with them right out of the box. I'd aim for cooking something fatty for the first few uses, but otherwise I wouldn't worry about it too much. Doing things like reducing acidic sauces (think tomato sauce, making marmalade etc.) will strip the seasoning quite a bit, but won't otherwise negatively impact your pan. If this happens, just fry some potatoes and carry on.