Should brake lever move if you press the (V-brake) calipers together?
I think you have the right idea. When you push the brake arms together, you are shortening the system, without changing the inner wire length. This in effect makes the inner wire 'longer', and because it is fixed at one end, it wants to move up towards the lever, exerting a force and causing the lever to move.
It seems likely that the low friction of the new housing/cable is helping to send all of the force to the lever end. I suspect that if there was more friction somewhere, you would see a different result, maybe the noodle would pop off the end of the housing, or the housing would come out of the lever at the barrel adjuster, but the housing would find some way to accommodate the relative increase in length of the inner cable, like this perhaps. At the moment, the lever itself is giving the least resistance and pivoting, perhaps this will always be the case.
The effect is further evidenced by you observation with the rear brake. Because the housing is interrupted and there is bare cable exposed, it is able to flex and bend into a curve so that the 'increased' length is accommodated and the cable slackens. This can actually be a useful aid if you want to disconnect a brake or derailleur temporarily, without undoing the cable pinch bolt. A kind of quick release.
On the front brake with uninterrupted housing, the cable is constrained by the housing and is unable to flex, so acts more like a rod, pushing the lever.
All of this, I think, is just to say 'yes' to your own observations and explanation, although I would be doubtful of its diagnostic merits, its just a fun thing you can note. Visual condition of the cable is more useful as well as the lever feel and responsiveness as you use the brake. these will degrade a little as the cable deteriorates.