I'm assuming your battery is the original, in which case it's time for a new one. This is common behavior for a rechargeable at the end of it's lifespan.
A blurred background is called a blurred background, or OOF background. Bokeh is the visual quality of said blurriness. "Good bokeh" means the blur is smooth, creamy looking, visually pleasing. A lens with less than good bokeh produces OOF backgrounds that look jagged, rough, basically not smooth and creamy, but unappealing visually. In general terms, more aperture blades = better bokeh. This can be seen with OOF lights in the background. A lens like the Canon nifty fifty has 6 blades, and OOF lights look hexogonal. The Sigma 50mm 1.4, OTOH, has 9 blades, and they're rounded, which gives OOF light sources a round, diffuse appearance. Any lens that produces excellent bokeh will have a higher number of blades.
The real source of your issues is the learning curve associated with using an SLR. You now have a body that is at it's best when you tell it what to do. Give it one wrong instruction, and you get something different than you expected. You gave a perfect example when you said you had your P&S set to auto ISO, but set your D40x to 100. Auto ISO lets the camera choose what it needs for shutter speed and ISO given your selected aperture, and it will generally give you a fast enough ss for handheld shots provided the light is decent enough. Manual ISO and aperture settings will allow the camera to choose only the ss, which, depending on light (such as indoor, like your shots) may not be remotely fast enough for handheld. This requires either a tripod or using flash for hand held. Increasing ISO will give you faster ss's, but at the expense of more noise. Start paying attention to all the numbers that display in your view finder when you half press the shutter button. They will tell you whether you should take the shot or not. Rule of thumb for handheld is shutter speed needs to be a minimum of 1/focal length to counter camera shake. For example, at 50mm, ss needs to be at least 1/50 of a second. A slower ss will result in a blurry or soft shot, as you've already seen.
Lastly, your max aperture has nothing to do with your camera. Aperture is a function of the lens. A camera can't set a wider aperture than the max aperture of the lens. You can't set aperture at 1.8 using a lens that has a max aperture of 3.5, for example.
In closing, read your manual. It's your best friend here. I realize the excitement of owning your first SLR and wanting to get right to work with it, but this ain't no P&S. I would also strongly suggest googling "Exposure Triangle". This will teach you how aperture, ss, and ISO are all related, and how they work together to give you proper exposure. I'm also going to recommend you purchase, or borrow from your library, "Understanding Exposure", by Bryan Peterson. You will benefit from reading it, and generally, you get much more positive feedback and answers on forums when you do your own homework. People get a little snarky when you want to be spoon fed, so if you can say "I read about the exposure triangle, but I don't understand x...", you'll get productive answers.
Keep shooting and have fun.