I have had a D200 with the 18-200 lens since it came out. Has always been more camera than I need, but it was the only one available (affordable for me) at the time that I could connect my gps to for location info.
I would like a body that does video that would work with the lens I have. Although, I am not opposed to getting another lens if needed as I will keep the D200, they just don't sell for much used. The kids love taking pictures and have an interest in learning the camera.
Most of our use is kid's soccer/basketball games, out on the boat, pictures of construction projects.
A couple of issues with the 18-200: 1) When the lens is pointed to the ground, the lens will extend on its on. 2) It looks like there are specs of dirt or dust inside the lens around the opening. The lens has never been detached from the body since purchase. I have read that dust will enter the lens when not attached the body.
Is there a repair/cleaning service for lenses?
I will not comment on Nikon bodies beyond the few comments below. While I do a lot of video I do not have the two latest Nikon "video bodies", which are the 3200 and the 5200. As I understand it the 5200 is Nikon's most friendly video camera, with various plug-ins and the best implemented live view and video features.
I have and use the D7000 which at the time of purchase was Nikon's best implementation of video. Frankly, Nikon is a weak second to Canon in video implementation to DSLRs.
If you are serious about video, for even the next few years, I would forget Nikon and go straight to a dedicated video machine or Canon.
DSLRs in video do bring a lightweight solution and a good look for docu and cinema, but the reasons for you are non-consequential, really.
Many using DSLRs want the shallow depth of field the lenses can bring, but you don't want that for sports, you want to set it on f/9 or f/11 and forget it.
One other thing Nikon bodies are terrible about AF motor noise being recorded. Sooooooooo, you will need an external mic, like a Rode, for example, that is if you want ambient sound at the game.
I do not own but have used the 18-200mm. It did not suit my needs for a quality image, but that is just me, do not follow my lead. For sports you will be f/7-11 anyway so my issue is not yours.
The two Nikon mount lenses I use are the Tokina 12-24 f/4 and the Sigma 70-200 f/2.8, on manual focus, exclusively. The two videos below used those two lenses and as you can see I (disappointingly) lost focus a couple of times on a bad weather, miserable day.
One last point, The further you zoom the more of a stable base you will need. A rock solid tripod with a decent fluid action panning head is your best bet. On the court you will be zoomed to 300mm, pretty quickly. Hand held will not cut it.
I will let others who have experience with the D3200 or the D5200 chime in on the video behavior of those bodies. I believe there are several owners of those bodies here.
Your best bet for a cleaning is to open a Nikon USA ticket - sending the lens to them.
Good luck. Let us know what you decide to do, and post some video. Ask more questions.