I don't know UK pricing these days, but That sounds like entry level territory to me. A few thoughts.
I understand that I need to raise the ISO to get a sharper image but
I got too much noise. That's why I am getting a DSLR with image
stabilization
Yes, this will be a huge step up, As, for the most part, the DSLRs shooting at its max ISO is going to have similar noise to a P&S shooting at 200-400, which is pretty low. The IS may or may not help. It can (and does) stop camera shake, but it doesn't do anything to stop movement in your subject. I've found that I can handhold (with a short lens) most of the stabilized systems down to about 1/15 of a second shutter speed, although at that speed, people can blur, even if they are just sitting.
I am using energy saving bulb in my living room
You might add more. I have a lamp from Target (big discount retailier in the states) that has 5 bulbs on individual goosenecks, where you can turn on 2, 3 or all 5 and aim them as you please. I bought it because it looks cool, but it is a pretty good photo tool too. It was about 30 dollars. Yes a good camera and lens help, but adding light to the room helps too. Also, consider an external flash. I'm not real keen on flash photos, but a good external flash that can bounce is a revelation. They aren't cheap, but a good flash with a pivot head gives very non flash looking flash pictures.
So my question is,
1. Do I need to get a fast aperture lens?
I sure would. Whatever system you buy, look for a 50 mm with a 1.7 or 1.8 aperture (varies by brand which they use). These lenses are fast, sharp, and cheap. Again, I don't know UK prices, but in the US, the nikon and canon ones are between 100-120 new, and less on the used market. I believe pentax has one in this range too. Sony doesn't (they only have a 1.4 which is relatively more expensive) but apparently minolta made a 1.7 and it is cheap and would work.
The fast aperture gets you faster shutter speeds which help not only with camera shake, but also subject movement. Downside is that you have less depth of field to work with, good for portraits, but a little trickier to focus.
This was shot with a nikon 50 1.8, wide open, at night in a pretty dim apartment with me on my belly. FWIW, I took a bunch to get a few useable ones. It helps to have a still cat...
2. Is that OK to use the bundled lens?
What's the suitable aperture range (with reasonable price)?. Most
lens come with kit are f3.5-5.6
Which is pretty slow. I'd get the kit lens because they are all at least passable, and tend to be useful focal lengths, and the bundled price usually makes them pretty cheap (I've seen some specials where the kit lens only added like 40 bucks over a body only price). They'll be good outdoors, harder to use indoors. I was shooting around the house the other day with my kit lens and no flash, just messing around, and was getting some usable shots at 3.5, but this was with a bunch of natural light coming in.
3. I assume I need to get a low shutter lag and fast focus DSLR?
The entry level ones are all pretty similar in this regard. The shutter lag is negligible on all of them (only noticeable when you compare to the pro cameras) and the AF is typically faster than most P&S cameras, though there is some variablility.
Do I have any choice with this price?
Sure. Pentax K100 comes closest to your original spec. They are pretty cheap, at least here, decent camera, have Shake reduction in the body. I can't tell if pentax has a 50 1.8 or not (B&H doesn't seem to have one) so that would be a limitation.
The sony A100 might also be a contender. Price keeps dropping, and it is a nice camera overall. You would need to look for a used minolta lens for a fast aperture on a budget, as the 1.4 version is around 300 dollars.
I'd consider the Canon rebel XT if you can find one. Doesn't have stabilization in body, but cheap, a good system, and their 50 1.8 is nice, cheap, and easy to get. Gut feeling is that these are discontinued (my store sold out and isn't getting any more it looks like) but I'm sure they are still around.
I'd skip nikon here, as much as it pains me. The d40 is a nice enough camera, but doesn't AF with any of their affordable fast lenses.