I'm assuming that I need to add something else into the LED portion to limit the voltage as well as current?
No, the resistor does that for you. Let's say you have 2V drop in the LED, then as a ballpark estimate the resistance should be (12V-2V)/10mA = 1k. 10mA is usually plenty to light an LED...use 4.7k if you really want 20mA.
I think you may find that the LED will not turn off when expected with 12V of reverse bias - many are only good for 5v.You may have to move it to the input side of the ULN2003.
don't forget to put a freewheeling diode on your relay's coil... Just to avoid ULN2003 frying...
Looking at the data sheet for the ULN2003A, it's rated at 0.5A for the collector current, so it shouldn't be a problem.
But you can't have them all pulling that at the same time. About 650mA is the total you can get away with so with 8 all on at the same time you are just about alright.
With the ULN2003, which is an open collector driver also with 2 added reverse bias diodes added for protection, I don't know if that will happen. Just don't be surprised if it is.