It's a bit like tightening a nut without a torque wrench.... do it up till it strips, then back it off a bit 

(ie, run the battery flat then you'll know....)

How about Wire Glue

picked up a Heathkit ET-3400;

Back in the day I had a Micro Professor which was a similar thing for the Z80 (Z80-based CP/M machines were in vogue). I got a few extras for it, like a Tiny-Basic chip.

Bloody thing got stolen from my office while I was on leave...

Here's a thought... on some makes of those long breadboards the left and right halves of the power rail are not connected to each other, and I see you are using both halves. Usually you can tell because the gap in the middle is wider than the other gaps, and yours look all the same, so maybe they are connected, but it's worth checking. You can easily bridge them.

The Fritz below shows what I mean.

I just divided 1000 by 60...

Hmmm, you'll need to explain that one to me Peter

Well, instead of turning on the LED, think that that pin could be the one that connects to the transistor and turns the transistor on and off.

So basically, to drive one motor, you could use the sketch that drives the LED, along with its switch, but amalgamated with the circuit of the transistor / motor to connect to that pin and Voila!

Then have a think about how to do it for two.

Edit.... This site is very good as a place to start on the electronics, especially of the components.

First have a look at this which will show you how to use a switch

Then, here you can see how to drive a dc motor.

Then think about how to put the two together, and how to double up for two switches and motors.

Well in spite of what the tutorial said, you should not run the servo from the Arduino's 5v, but rather from its own external source.

It might be that a peculiarity in the 4th motor is causing it to draw too much current which the Arduino cannot manage. Lots of servo funnies are solved merely by providing external power. If that doesn't sort it out, then further discussion is needed, but I'd certainly try that first.

Long as you check the voltage required and current that is drawn by your motors- you'll need to check the datasheet of the motors.

I make that almost one order of magnitude out - more like 1/17th of a degree

arctan 1/1000 = 0.058 = 1/17 yep....

But that's when the beam is normal to a plane. Sketch attached, which I drew with a 30 degree angle since all I could find in the house was a zillion set-squares and no protractor, shows the angle produces a much magnified distance on a plane at greater angles.

And just for the hell of it, I attached the Fritz of my set-up.

It is indeed a bit messy  and difficult to follow the wiring in your photo... have you double-checked that all the grounds are connected as mentioned and shown in that link- battery / servo / Arduino / TLC chip all need common ground.

I can't make out your wiring

I've got one of these from DFRobot, see below. The servos are controlled by a 2-axis joystick using a modifed version of knob, and the joystick button switches trhe laser.

Are you projecting the laser onto a plane? Because if you are, remember that the change in angle will not bring about the same linear change along the plane as the angle varies and to get your 1mm accuracy you'll need to calculate what that change in angle will be as the angle itself changes.

Also when the beam isn't normal, you'll get an oval dot and that might be construed as inaccurate....

Surely somewhere UK will have a board?- maybe not the identical brand, but similar 298 based one? I don't know what HM's Royal Mail is like nowadays, but someone should be able to get a board to you overnight...

Maybe you have the external LED in back-to-front?

This whole pullup/pulldown nonsense makes a lot more sense now

Taking the non out of non-sense...