1a. What's the best way to contact a major team of developers for projects under contract, or at least at a point where you can determine what the budget should be for that particular game while using that specific development team?
1b. Should if be the team's project manager that I contact, a desk associate, or someone else?
1c. How should I contact them; by phone or email?
2. What should I expect and look for in a team besides their work history and portfolio? I know contacting a large studio like Volition is highly unlikely, but what about a mid size one like Dark Side Games? What would they expect me to have upon my presentation beyond a product outlook, profit expectancy, market value analytics (if necessary), etc.?
3. Finally, are there any best practices I should use to stay in good favor with my prospects should I not have everything I need upon my first pitch or presentation?

1a-c. Find a contact phone number. Usually it is on their web site if they are serious about contract work. Initial conversations in the business world are still primarily handled by phone, then transition to email. They go by various titles, "development manager", "account manager", "project coordinator", the business owner, VP of whatever, and more. Eventually you'll likely be working with a "producer" on their end, but that is after the business contract stuff is taken care of. Usually just follow the "Contact Us" button, it will have a phone number, call during business hours. You'll get a secretary, briefly state that you are looking to discuss a new contract and would like to speak with someone about that, and they'll transfer you.

After a small number of phone contacts it will quickly move to emails. Then it will move to discussions with your business lawyer to draw up the contracts. There will likely be one or more in-person meetings, on the last one contracts will get signed, which is frequently a financial milestone. With luck the lawyers will never be involved again because the business relationship will be amazing, hopefully.

Be prepared with budget numbers in advance. If you're approaching a development house have your budget known in advance, is it a $200K quick project, a $2M longer project, a $20M large project? If you are approaching individuals, are you willing to pay at a low-end $50/hr with milestone completion bonuses, or a more professional pay rate of $150 or so per hour plus milestones and certain expenses? (Or perhaps you are looking to outsource to Pakistan or Russia or Peru or someplace cheaper than these rates?)


2. Work history and portfolio is mostly it, but if you have other business contacts you can ask for recommendations for that type of game and you can ask the company for references. You want people or a company who has developed similar games under contract before. You mention Volition, they are not really a contract development studio. Exactly what type of business or individual you want depends on your needs and desired game. Sometimes you can hire a single individual, sometimes you need to hire a larger team, sometimes you need to contract out to a moderately large business. Cost scales accordingly. They typically do not care about YOUR profit expectancy for the product, YOUR product's market value analytics; they care about ensuring they can make the product agreed to by the contract and verifying that you will be able to pay the bills at the milestones.

3. Stay in contact. Be open about both the good and the bad. You will be working with them a lot. At first you will work closely with them to establish all the fine details. Many meetings will be at the beginning establishing what exactly they will be doing and what you will be doing, what you will be providing, what you will be expecting, how it will be delivered. Usually fewer meetings and contacts in the middle, perhaps down to twice weekly or once weekly, but I wouldn't go that low unless you have a strong existing relationship as too much can go wrong. At the end it will move back toward daily, then near constant communication as you finish up. Depending on both your and their location on the globe you may have multiple in-person visits as well.

/edit: Ninja'd by Tom as I had to take my kids to school in the middle of the reply.

What you're looking for is their business development managers (BDMs): their salesmen.

What you need is to create an RFP or seek partnership (I recommend the latter).

I suggest you present yourself at a game convention (not E3, preferably) such as GDC or the likes, and find the booth or a representative of a studio that does servicing work.

Your current post does not make it clear whether you'd like to have creative direction over development. Are you seeking to outsource programming alone? If so, it would suggest you'd have some kind of GDD in hand and a few influences/references they could look into. It's often good/easy to start a relationship by saying something like: "This is going to be a match-3 puzzle game with some RPG elements" (although, please don't go down that route, the market is over-crowded with these nowadays).