Relay Circuit – Replacing LEDs with Diodes

First, get yourself a cheap DMM.

Next, measure the 5 volts on the 2003 board, between V- and V+ (are you sure you connected the V- to your 5v ground?)

If you've got a good 5 volts, start measuring your control lines. If all 4 are zero (or less than about 0.5 volts), that will explain why your LEDs aren't lighting.

Best practice:

1. (per @PlasmaHH's suggestion below) Safety. If a state is unsafe, make it not the default;
2. low power consumption. If you know beforehand that relay will be in specific state for longer time, it is wise to design the circuit so that this state will be obtained within 'off' state of the relay (where there's no, or minimal current is flowing through its control element);
3. mechanical fatigue. For mechanical elements of the relay, there could be switch position when relay's internal mechanical elements are in 'strained' mode - it is wise to minimize the time relay elements are in this state.

Now about your choice. You can see that those 8 relays are two position ones, in 'off' mode central pad is connected to one pad, in 'on' mode to another, thus you have a freedom of choice connecting load to switching terminals.

should I be using NC here?

NC is usually stands for 'not connected', so you should not use it. As @Tyler said below, NC is 'normally closed' or 'normally connected' and NO is 'normally open', details can be found here in Wikipedia. But to be precise you have to refer to device's datasheet.

The other bit which I am unsure about is....when my ESP8266 is booting up, is there a chance that my relay control pin will go LOW before i want it to?

To answer this question circuit diagram is needed. The page you linked to is about turning electric lamp on/off, you seem to talk about powering Arudino using the relay. Is it about self-power off?