Can you use a flash?
If so, then motion blur shouldn't be a problem if you're within the flash range indoors (because the subject would not be properly exposed except during the flash burst, and the flash burst is so short it can freeze the action).
If you can't use a flash and stay within the rated flash range, then it's a another ballgame (most cameras won't work well enough without a flash in most indoor condtions if you're taking photos of a moving subject).
Your Fuji is going to be especially bad in this area (without a flash), since it's limited to ISO 100. Almost any popular camera can go to ISO 200 or more. Each time you double the ISO speed (which represents how sensitive the sensor is to light), the camera can use shutter speeds twice as fast for the same lighting and aperture.
So, most cameras are going to let you get shutter speeds at least twice as fast as your Fuji in low light.
Your Fuji also loses a lot of light as you zoom in more. It's down to a maximum aperture of f/4.8 on it's long end (most optical zoom). Smaller f/stop numbers are better for light gathering ability.
So, yes, almost any camera would be an improvement over your Fuji.
From your descripton of problems, it also sounds like you want something that focuses faster. If you read the Conclusion section (last page in each review before the sample images), you'll see comments on Autofocus Speed and reliability.
But, for condtions like Basketball in typical gym lighting without a flash, you will need to step up to a DSLR model (and you'll need a bright lens to go with one) if you don't want any motion blur. None of the non-DSLR models are going to handle rapidly moving subjects in low light without a flash (and even a DSLR is going to be challenged in those conditions).
Condtions like that usually require ISO 1600 *and* a very bright lens if you want to get rid of motion blur (and most non-DSLR models will have unacceptable noise levels at much over ISO 200 or 400).
Even in typical room lighting at night, it's tough to get photos of a non-stationary subject without a flash if you don't want any motion blur, even with a DSLR using a bright lens (even if they're only walking).
What's bright to the human eye is not to a camera's lens. What you think is a well lit interior is very low light to most cameras. That's where a flash comes in. ;-)
Out of the models you're looking at, I'd probably lean towards the Panasonic or Canon models. But, I wouldn't expect to get a high percentage of usable images in many low light condtions without a flash indoors if your subjects are not stationary.
For that, you'd want a DSLR capable of using much higher ISO speeds along with a bright lens (which increases size, weight and cost).
With an ultra-zoom model, in order to give you that much optical zoom in a very small package, the sensors have to be very small, too. Because they're very small, they have a smaller surface area for each pixel. As a result, they don't generate a signal that's as strong, requiring more amplifcation (which adds noise/grain) for equivalent senstivity to light. That's one of the tradeoffs with smaller cameras.
In contrast, a DSLR uses a much larger sensor, and can give you much higher usable ISO speeds (which you need to get shutter speeds faster to help with motion blur). But, they will also require much larger and heavier lenses for equivalent focal range (how much magnification from wide to long) and brightness.
There are some exceptions. For example, Fuji has managed to get relatively good high ISO performance in a couple of their new subcompact models (like the F10 and F11). But, they're not ultrazoom models, and they lose a lot of light as you zoom in more.
Any camera choice is a compromise.