Getting a 70-200 VRII and using it on the D7000 will surprise you how much of a difference a lens can make. Sure, getting a D800 or D4 with a 70-200 will give better performance than a D7000 and 18-105 but a D7000 with a 70-200 will be better than a D800 with a 18-105. You are currently holding back the camera you have although as a general purpose lens the 18-105 is better than most people assume due to its kit status. So the we get the heart of it, you want a new camera and it is not for a specific need or deficiency of the current one. That is very common and is a fine reason to get a different camera. My only concern was that you were assuming the upgrade would change the look of the images and finding it did not, would be disappointed. Note that for the same field of view of a 70-200 on a D7000 you will need a much more expensive 300 2.8 for the field sports. Since you are seeking memory capture, and not fine art, I would suggest considering a camera that will be with you most of the time to capture unplanned events as most personal occasions in the life of a child are. You ARE going to miss most of those shots with a D800 and pro glass, more than the D7000 and still less than if getting something like a V2 that would be with you 90% of the time while the D800 sits in the bag back home any time a photo-excursion is not planned. I carry my backpack with me a lot but even with only 2 lenses it it, it gets taken out a lot less than when I was primarily using a D90 and 18-105 because of bulk and weight. The weight and time to set up is so different that I took about 10 times as many shots with the D90 per month than the D800. I have just under 100,000 D90 shots recorded in the same span of time I have had the D800 which has about 10k frames through it. Shooting with the D800 is more deliberate, slower in pace and more selective in subject and much less spontaneous.
I have more memory, chance, captures on my Galaxy SIII phone than either the D7000 or D800 because the cameras coupled with pro lenses is just not the size and weight to carry casually and into placed where security is an issue, such as leaving the bag in a cloakroom in a restaurant or theater. Lugging 35lbs everywhere that is needed to cover the same range that a V2 does at 10 oz really changes how they are used. A good example. I shot a wedding Friday and used two bodies and some good lenses. The portion that was planned was covered very nicely by the D800 but in the limousine that took part of the party on the traditional drive to various monuments to place flowers, my phone camera alone could get the candid opportunity shots that were spontaneous chance situations that will be family folklore for decades to come. What is the best camera....the one you have with you. If you are older, as I assume, how often are you going to be lugging a pro camera and its lenses around unless a dedicated planned photo excursion? These points are brought up because they are often not considered when thinking of a shiny new camera. It will be counter productive if you do not carry it everywhere that you carry a light compact camera.
If you really want a D800 or D4 and the major investment in lenses and tripod, plan on a good mirrorless shirt pocket camera as well, because you will capture more of the unplanned shots with it. The V1 or 2 will work with your pro Nikon lenses with the adapter. Remember that memory prints are not any more valued or dismissed based on ultimate image quality. Having the print is all anyone cares about. They are seldom printed large. Stan St Petersburg Russia