The Fuji 5200 does not have image stabilization. But, it does have higher ISO speeds available compared to your Z1.
As for the Nikon D50, forget the 18-55mm kit lens for indoor use, unless you plan on using a flash. You might be able to get away with using it in some lighitng if you had relatively still subjects. But, it's not that bright of a lens for this purpose.
I'd get one for outdoor use, or indoor use with flash anyway, but you'll need something brighter for existing light shooting without a flash.
You'd give up the advantages of higher ISO speeds quickly if you zoomed in much with the kit lens.
The lens on your Z1 will be about 4 times as bright as the 18-55mm f/3.5-5.6 kit lens if you zoom in much with the kit lens. So, you'd need to use ISO 1600 on the D50 to get shutter speeds as fast as you could with ISO 400 on your Z1.
The Z1 lens can maintain f/2.8 throughout most of it's focal range, only losing 1/2 stop to f/3.5 on the long end (which is DRAMATICALLY longer than the long end of the Nikkor kit lens).
The kit lens will drop off to a maximum available aperture of f/5.6 if you zoom in much with it (and the f/2.8 you'd have with your Z1 at equivalent focal lengths is 4 times as bright as f/5.6)
So, get yourself a bright prime like a 50mm f/1.8 if you're on a budget. It may not be just right for focal length purposes (it could be too long in some areas), and you'd need to use your feet for zoom in larger areas if it's not long enough. But, it's bright, sharp and inexpensive.
Then, as budget permits, get yourself a wider prime for when you need it for closer quarters so that you can get a bit more of a scene in the frame (you can only back up so much). ;-) Something like a 28 or 35mm are popular choices for indoor use without a flash in close quarters.
You may also want to look at third party choices like the Sigma 30mm f/1.4. f/1.4 is but twice as bright as f/2, 4 times as bright as f/2.8, 8 tmes as bright as f/4 and 16 times as bright as f/5.6
If you want a zoom lens, make sure you get one with f/2.8 available throughout the focal range (f/2.8 is the brightest zoom you can get in most camera mounts) for indoor use without a flash.
Larger Apertures = Smaller f/stop numbers = more light gets through = faster shutter speeds for the same lighting and ISO speed. A brighter lens adds size, weight and cost. But, the "kit lens" just isn't going to cut it for kids running around indoors if you can't use a flash. Even a bright prime will have limitations indoors.
Also, unless you've got some experience with an SLR shooting at larger apetures, you'll have a learning curve.
Depth of Field (the amount of the scene that's acceptably sharp as you get closer or further away to the camera compared to what you're focusing on) will be very shallow at larger apertures with a DSLR. A non-DSLR model will have dramatically greater depth of field for any given aperture, 35mm equiovalent focal length and focus distance.
So, you'll need to learn how to cope with a shallow depth of field, especially at closer ranges to your subject, in order to achieve the desired results shooting without a flash indoors.