When do sieges start

A true minimalistic build is 1 anvil and 2 copper nuggets; so I'm going to assume that's what you meant.

1. De-construct your wagon (you have no other way to get wood).
2. Make your wood furnace (copper nugget #1)
3. Cook 1 ash and 2 coal from your three wood
4. De-construct your furnace
5. Use the ash to make a smelter (ash is a fire safe material in DF)
6. Smelt 1 copper bar (you're going to need the other one)
7. Build a forge (nugget #2 + anvil)
8. Forge a battle axe (use this to cut down 3 trees)
9. De-construct your forge
10. Build a second wood furnace (with the nugget from the forge)
11. Bake 1 ash and 2 coal again
12. De-construct the furnace
13. Build a forge out of the ash and the anvil (again the ash is a fire safe material)
14. Smelt the copper bar from the wood furnace
15. Forge a pick

This gets you a forge, a smelter, an axe and a pick, thus completing the minimalist challenge.

Or you could just start out with 2 ash, 2 coal, an axe, a pick and an anvil and skip this otherwise tedious beginning.

Its worth noting that with 1 copper nugget you have the choice of either a pick or an axe. As a result you either don't have enough copper or not enough wood to complete the challenge.

Edit: actually if you had above ground lava, you could avoid needing the second copper nugget and make two ash giving you have enough for the forge (as lava forges don't need coals).

DF2010

In DF2010 (the latest build) you can use the training axe in place of a normal axe to cut down more trees. People who traditionally used the minimalistic build consider this "cheating" as you essentially get your axe for free. This can let you use 1 copper nugget.

1. De-construct your wagon (you have no other way to get wood).
2. Build a carpenter's shop (wood #1)
3. Construct a training axe (wood #2)
4. Make your wood furnace out of the copper nugget
5. Cut down 3 trees
6. Cook 2 ash and 2 coal
7. Use the ash to make a smelter
8. Deconstruct the furnace to get your nugget back
9. Smelt your copper bar (using ash #1 and the nugget)
10. Build a forge from the ash and anvil (using ash #2)
11. Forge the bar into a pick

This gets you your forge, smelter, axe and pick. Everything you need to build a full society.

The wiki doesn't know, so it's safe to say that that is one of Toady's secrets.

After being sufficiently "successful" at defending against the siege (killing sufficient attackers, waiting them out, or some combination thereof), the attackers will retreat. All of the remaining squads and groups will head for the map edges and leave, typically favoring the edge they entered from. Once the last of the siegers leave, the "SIEGE" tag will go away.

Source: http://df.magmawiki.com/index.php/Siege

However, the 40d page states that if you do nothing, it will take a very long time for the siege to end.

The siege will end after some (LONG) time have passed,

Source: http://df.magmawiki.com/index.php/40d:Siege