Does the Siege-Gang Commander have haste?
Only creatures in play when Burst of Speed resolves are affected - it even says so in the rulings. But in your example, it won't matter because by the time Goblin Lackey lets you get your extra Goblin out, you're already well past your chance to attack (unless you somehow get a second combat phase).
If you're curious about the actual rules behind this, unfortunately, it's one of the more annoying nitpicky details. Spells and non-static abilities which modify characteristics (name, mana cost, color, color indicator, card type, subtype, supertype, rules text, abilities, power, toughness, loyalty, hand modifier, and life modifier) determine the set of objects they apply to when they resolve. Haste is an ability, which is a characteristic, so Burst of Speed only affects the creatures that are on the battlefield when it resolves.
On the other hand, spells and non-static abilities which don't modify characteristics are considered to be modifying the rules of the game and determine what they apply to continuously. For example, there's Dread Charge, which looks like it's granting something like fear, but since it's not actually granting an ability, will indeed apply to creatures which come into play after you cast it. Alternatively, in the context of your example, if hypothetically haste weren't an ability, and instead Burst of Speed said "Until end of turn, creatures you control can attack even if they haven't been under your control continuously since your most recent turn began", then it would affect subsequent creatures that enter the battlefield.
"Indestructible" was recently "fixed" to become a keyword ability, so like with haste and Burst of Speed, for example, Boros Charm now only applies to creatures in play when you cast it. On the other hand, "is unblockable" was changed to "can't be blocked" to emphasize that it's not an ability, so for example Glaring Spotlight's ability does apply to creatures which come into play after you use it.
Here's the full rule:
611.2c If a continuous effect generated by the resolution of a spell or ability modifies the characteristics or changes the controller of any objects, the set of objects it affects is determined when that continuous effect begins. After that point, the set won't change. (Note that this works differently than a continuous effect from a static ability.) A continuous effect generated by the resolution of a spell or ability that doesn't modify the characteristics or change the controller of any objects modifies the rules of the game, so it can affect objects that weren't affected when that continuous effect began. If a single continuous effect has parts that modify the characteristics or changes the controller of any objects and other parts that don't, the set of objects each part applies to is determined independently.
Example: An effect that reads "All white creatures get +1/+1 until end of turn" gives the bonus to all permanents that are white creatures when the spell or ability resolves -- even if they change color later -- and doesn't affect those that enter the battlefield or turn white afterward.
Example: An effect that reads "Prevent all damage creatures would deal this turn" doesn't modify any object's characteristics, so it's modifying the rules of the game. That means the effect will apply even to damage from creatures that weren't on the battlefield when the continuous effect began. It also affects damage from permanents that become creatures later in the turn.
Finally, static abilities also apply continuously. This isn't usually confusing; they're written as simple statements of fact, like Blur Sliver's "Sliver creatures you control have haste", which while in play gives all your Slivers haste, plain and simple. The full rule for that:
112.3d Static abilities are written as statements. They're simply true. Static abilities create continuous effects which are active while the permanent with the ability is on the battlefield and has the ability, or while the object with the ability is in the appropriate zone. See rule 604, "Handling Static Abilities."
No, Burst of Speed only affects creatures in play at the time it resolves.
From the ruling on Burst of Speed at the bottom of the page:
10/1/2009 Burst of Speed affects only creatures you control at the time it resolves. It won't affect creatures that come under your control later in the turn.