That's a lot of questions, and most of the answers are "it depends", or there's more than one right answer.

I'd suggest you consider reading The Art of the Start[1] by Guy Kawasaki, The Four Steps To The Epiphany[2] by Steve Blank, and High Tech Startup[3] by John Nesheim. Between those, they cover a big chunk of the basic stuff you need to know.

The Founder's Dilemmas: Anticipating and Avoiding the Pitfalls That Can Sink a Startup[4] also has a good reputation, but I haven't had time to read it yet, so I can't give a personal endorsement. But it sounds like it might be worthwhile.

FWIW, though, I can tell you what we did at Fogbeam[5]: The company is organized as a legal entity, but we are an LLC right now, not a Corporation. I would not necessarily recommend doing that to anybody else though... while it is possible to build a big company as an LLC, practically speaking, if the intent is to build a scalable startup, the kind that's going to seek VC money and that you ultimately hope to IPO, you need to be a corporation. VC's essentially will never invest in an LLC (there are technical reasons why) and LLC's have serious limits if you need more than a hundred or so "shareholders" (I forget the exact number, but it's a pretty small number). The reason we are an LLC is a historical coincidence, rooted in weird shit that happened back when I was planning to start a consulting company, before deciding to do a product. Fortunately the prevailing wisdom is that it's fairly straightforward to convert from an LLC to a C Corporation. When they day comes that we need to do it, we'll switch.

As far as equity distribution, we have been operating on a handshake agreement between the founders. Technically speaking, I own 100% of the company "on paper" since I'm the only Member listed in the LLC operating papers. But that is, again, only a historical legacy. I created the company and worked alone for the first year before inviting the first co-founder onboard. It would be easy enough to amend the papers to update the equity split, but we've basically taken the approach that "well do that when we convert to a C corp, or there's a specific need to" (like, if we get an acquisition offer). So, yeah, we are operating on trust at the moment. A lot of people will recommend against doing that sort of thing for various reasons (see: The Social Network for example), and I have known friends who got screwed by co-founder disputes because things weren't put into writing up-front. If I had to advise somebody, I'd probably advise you to decide on the equity splits, intellectual property assignments, etc. up-front, and formalize everything from the beginning just to be on the safe side. The downside to that is, it costs a little bit of money and time. shrug

As for YC... nothing against them, but I'd never make a decision based on "what YC wants". But I look at all accelerators / incubators / etc. as "something that might be nice to do, but we'll succeed with or without them." Being that we are an East Coast startup with constraints that would limit out ability to move to CA in order to do YC, we've never even applied and probably never will. If doing YC is super important to you, then maybe you should treat this a bit differently. It's up to you.

Design document? Meh... I mean, yeah, but no. Not exactly. You need to know something about what you're building, but as a rule, you probably won't know exactly what you really need to build to achieve "product / market fit"[6] right away. So no sense spending months on an elaborate BDUF design doc. Sketch out the high level design, and IF you wind up subcontracting any work, then you'll have to formalize a spec for the contractor. But don't spend months and months designing something nobody wants. "Get out of the building" as they say, talk to customers, and iterate the design as you learn what people want/need.

[1]: http://www.amazon.com/Art-Start-Time-Tested-Battle-Hardened-...

[2]: http://www.amazon.com/Four-Steps-Epiphany-Steve-Blank/dp/098...

[3]: http://www.amazon.com/High-Tech-Start-Revised-Updated/dp/068...

[4]: http://www.amazon.com/The-Founders-Dilemmas-Anticipating-Ent...

[5]: http://www.fogbeam.com

[6]: http://www.stanford.edu/class/ee204/ProductMarketFit.html