Does the screen resolution override the source resolution?
They both matter.  What really matters is the combination of the two.  Scaling always produces a loss of quality unless you are using an even multiple of the resolution.  This is because interpolation must occur during a scaling operation that isn't an even multiple or divisor of the original resolution.  This is due to the fact that some pixels must contribute to two different new pixels.

That said, it is even more complicated when deciding what to shoot with for your camera if it is coming from a camera.  The actual imaging sensor has a native resolution itself and will internally apply scaling to fit to whatever size you need.  In some cases, like DSLRs, all formats actually end up getting scaling applied since the actual sensor is much higher quality, but that also means the down-scaling can be more accurate if implemented properly.

If you know that you will only ever use output in a given resolution, start with that resolution as early as possible to make use of any source side optimizations that may exist rather than going to an unnecessary intermediate format that will only be scaled again (resulting in another interpolation).

If you are targeting multiple output formats, then use whatever the highest quality format you can is (even if it exceeds the maximum format you intend to output) since the more information you have, the more accurately you can downscale later.

The exception to this would be if the majority of your usage will be on the highest quality format that you will be distributing, then go ahead and use that quality for source since you will gain a little quality at that level at a cost of a little quality on the lower end videos.

As for the side question about screen size, yes, it can make a difference, but angle of view is what matters.  A 50 foot screen two miles away is still going to have less visible resolution than a similar pixel count screen that is 4 inches across and 3 inches in front of your face.  I touch on this in a little more detail in this answer.