If the NID isn't in a problem area it is unlike the NID will be moved by Verizon. It needs to be located near your electrical ground since the NID itself ties into the ground. As such, I wouldn't move it for risk of violating a wiring code or getting Verizon upset since it does belong to them as a demarc between their responsibility and your responsibility. NIDs can be removed from the house temporarily if things such as siding or remodeling is done but moves typically require them to come out and do, not that you can't of course!
See what you can do for us as far as finding out your statistics. I'm confused about what you meant in the last section of your post, but if what you meant was a matter of tying the modem and the house line directly into a "split" of the feed from the NID, that's something I would avoid doing. The NID test only takes a few minutes and it will knock out your phone service as a phone will not be connected (as such, line will keep ringing), but that is what we want. If we can get just the modem online with no additional home wiring, preferably using the modem's supplied cable since it is a quality cable, that would rule out any issue inside of the house. Introducing flat extension cable or home wiring may skew the stats off. No way for sure to know this from a forum post but it never hurts to be on the safe side (which is why I post cautiously!).
If you have an older style NID without the capability of using a test jack, and instead one of the older ones that uses screws, bolts, or other tie-down/punch methods, you will need to consider disconnecting your home manually to fully test. Verizon can replace this for you with a newer box to make things easier, but as long as you connect the modem somehow to it (DSL modems don't care too much about polarity, so what is tip and what is ring doesn't really matter here. That's mainly an issue with phones), and then re-connect your home exactly the way it was before (each wire in it's proper connection as found) you're in good shape. The newer plastic NIDs are the easy ones as they simply disconnect the house when the test jack is revealed, or if it's like my NID, you disconnect the house by unplugging the RJ-11 jack from the test jack and then you're good to go for testing. Some NIDs kill the house as soon as you plug something into the jack as well.
