What's a good way to judge the number of hatcheries that I should have at any point?

So there are really three questions here and while they are related they really feel separate. I'm willing to supply three answers here, but in the future you should seriously consider splitting them up.

Do I need a second hatchery at my first base before I go to Lair?

3 Hatch on 2 base is actually a very new build that is just starting to become standard. Before that there were a number of Korean players who played around with early aggression based on 2 Hatch on 1 base with heavy Zergling play.

The important thing to consider when considering having more Hatcheries than you have bases is:

• Do you need more larva?
• Can you get this larva another way?

If you are constantly starved for larva (as many Zergling builds are because of how cheap they are), then you need more larva. The more cost effective solution is additional queens, but lacking that (ie. full queen saturation) a Hatchery does help.

By comparison builds which production is bottle necked by other factors (Roach/Hydra, Ultralisk or Broodlord centric) requires less larva, because your bottleneck is elsewhere.

If you're trying to decide how many Hatcheries to build the most important feature is: What am I trying to build? Based on that you can decide how many Hatcheries you need.

Now if its not obvious to you when you need more larva, people have created tools to help.

Typically, do the Zerg expand before Protoss and Terran or is it roughly the same?

While most players do choose to expand that is usually a personal choice and related to their build.

Zerg have a number of things which make expanding easy:

• Cheap Hatcheries (350 including drone which is usually amortized over its lifespan)
• Mobile force (Zerglings are one of the fastest units in the game, and Mutalisks tend to lend themselves to map control)
• Movable defensive structures (both Spine and Spore Crawlers can be uprooted to new places as you expand).

Conventional philosophy dictates that a Zerg player should have 1 more expansion than the equivalent Protoss or Terran player, but that is not a hard and fast rule, only a convention.

As for when to expand... map control helps...

How important is the role of the queen in boosting production? Is that a central theme, or should I just be adding hatcheries?

Queens produce 4 larva every 40 seconds, or 1 larva per 10 seconds.
A Hatchery produces 1 larva every 15 seconds or 3 larva per 45 seconds.

Queens produce larva 33.33~% faster than a Hatchery. A Hatchery produces larva 50% slower.

A queen is half the price of a Hatchery, so we can say a Queen in ~66% more efficient.

That said, many people have a hard time being perfect on their queen timing (I have problems with it myself) as a result they may have a backlog of energy. In cases such as this, adding on an extra Hatchery will increase the effectiveness of the Queens you currently have.

Alternately, many builds rely on building more Queens than they have Hatcheries (for spreading creep or AA). Once these requirements have disappeared these Queens are idle. In the case of idle Queens the production of a Hatchery is very efficient.

As a player of Zerg myself, I can give you some advice.

Q1. Do I need a second hatchery at my first base before I goto Lair?

Not necessarily, it always depends for what build your going (what is your first objective).

Q2. Typically, do the Zerg expand before Protoss and Terran or is it roughly the same?

Here you have to keep in mind that your hatcheries are your only production building. They produce drones which makes your economy and your army forces. In short, the answer is yes. Keep in mind that a Zerg usually need 1 more base than the opponent.

Q3. How important is the role of the queen in boosting production? Is that a central theme, or should I just be adding hatcheries?

That's another big question. Keep in mind that each queen you build is half an hatchery. Queen is 150 minerals and 300 is a hatch. Queens are usefull for a lot of reasons. They spread creep, they can defend early push and they can oversaturate you larva production.

It's always up to you, depending on the game, to decide if you need a queen or another hatch. Usually, they are very usefull all game long, but make sure to keep in mind that it's a lot of minerals.