Molex power with USB serial Communication from the same PSU
Two important things first:


The Uno 5 V pin and power bus must never be supplied more than 5.5 V or the MCU may die. The DC power input jack (the barrel jack) and the Vin pin, which are connected together, are not directly connected this bus, which is why you can supply higher voltages to that (see below).
You can not simultaneously supply power via both the USB cable and a power supply to the 5V pin/5 V bus because that would directly connect the two power supplies. That's a very bad thing, and probably not just for the Uno. (But there is a trick you can use here to still connect USB data when you are supplying power to 5V; see below.)


Now on to the details.

You can safely connect Vin (the pin or the barrel jack; they are directly connected) and USB at the same time because there's a switch (T1 on the schematic) on the board that will disconnect the USB power line from the Arduino 5V bus if more than 3.3 V is being supplied to Vin.

However, Vin goes through a regulator which, while it has the advantage of safely handling voltages much higher than 5 V, also causes some voltage drop, so you need to supply it with a minimum of 7 V. (6 V might work if there's very little power draw; 5 V almost certainly will not.) The maximum voltage you should supply is 20 V, but it's recommended that you stick to 12 V or less, since higher voltages mean that the regulator can get rather hot.

Unfortunately for you, this doesn't really help with your problem. While you could take the 12 V power off the molex connector, supply it to Vin and run the Arduino itself quite happily with the USB cable also connected, the regulator on Vin is rated for only 1 A, which is considerably less than you need for the entire system.

There are ways to have some parts of the system powered by one power supply and other parts powered by a separate power supply, but they're not trivial (and also beyond my expertise at the moment :-)).

Probably the easiest solution is to connect the 5 V output from the molex connector to your system power bus, which would be supplying both the 5V pin on the Arduino and your external components, and then build a custom USB cable with only the D+ and D- pins connected, dropping the power and ground pins. (You don't need the ground pin because the USB data signal is actually a differential signal: the difference or similarity in the voltage between D+ and D- pins is what determines whether the signal is a 1 or a 0.)

By the way, just to make sure, you do have some separate switching for all those LEDs, right? You're not trying to provide all their power via the Arduino digital outputs? Because, as per section 32.1 of the ATmega328P datasheet, while you are allowed to sink up to 40 mA from any individual data pin, you cannot sink more than 200 mA in total from the chip.