>The main thing I find is my camera tends to blur whatever isn't the immediate focus of the picture. Is this to do with the autofocus like I suspect, and what can I do to fix it?
This part is a little complicated.
The region that is in focus with an optical lens, is a spherical shell from the eyeball outward. Everything within 2 to 3 feet being in focus is a common effect.The width of this region is called the depth of field. You want more DoF and you have a narrow one now.
The range is defined by the lens aperture, which you've got at 3.5, which means wide-open. It's good for dark rooms like that to be more wide open. But then you lose the ability to far-focus.
I'm afraid there is no fix. What you must do is use a longer exposure , or brighter light. They both mean more light. In the sunlight you can use f/8 and get more stuff in focus.
Indoors like this, you should use a tripod and long exposure. maybe just setting the camera on something sturdy and clicking timer mode will get a stationary shot. That way you can expose for as long as you want, use whatever aperture you want (f/6.3 or f/8 would be better for your purposes) and get a clean shot (lower ISO).
If you cant use long exposure because the subject moves (people or animals) then we're stuck with making the light brighter. Big flashes. But now youre seeing by the light of the flash. Youll want one bigger than a quarter inch.
hard edged shadows are caused by point light sources. If the ceiling was all white, there would be no hard shadowing. That's why I like to point my flash at the ceiling and light the room instead of the subject.