@ thalassic: I'm so sorry to hear about your cat
13 isn't a bad age really, but we sort of expect more these days.
@ wolf: 6 weeks is very young for your kitten to be away from his mother, 12 weeks is the recommended age now. Advice? Don't change his diet for a good while, get him jabbed as soon as you can, and keep a sharp eye on him for the next month - if he shows any signs of lethargy, runny nose, weepy eyes get him straight to the vet. If he is a flu carrier this time of stress at changing homes will be when it comes out and he could become a very sick little kitten very quickly (and I don't want to frighten you but cat flu is a killer in young kittens). Keep him contained to one room to start with and slowly introduce him to new bits of the house - don't just give him the run of the place straight away, he'll get lost and you'll lose him! Don't put his litter tray next to his feeding area, and separate his water and food by some fair distance - cats don't like eating where they drink. Don't expect him to ever use the lovely cat bed you've no doubt bought him! If you have any breakable ornaments put them away, and kitten proof the same way you would baby proof. I have genuinely had kittens climbing the wallpaper behind a big Welsh dresser. They made it all the way to the top, a good 8 feet up, and then couldn't get down...
As for introducing the dog and cat, this is a long slow process, it isn't likely to happen instantly, you will need to work at it. You don't say what sort of dog it is, but the kitten is tiny and fair prey for just about any size of dog. Make sure the dog really does understand NO! have the kitten on someone's lap, and bring the dog in on a lead, handled by someone who isn't afraid to reprimand the dog. let the dog approach the kitten and sniff, but at the slightest sign of either over enthusiasm or aggression from the dog, or fear from the cat separate them, and tell the dog off. The worst thing you can do is let the kitten run away, the dog's natural instinct is to chase which will lead to tears. If they react sensibly and calmly praise them both excessively. Leave it at that for the first session. Repeat daily for as long as it takes. Obviously the dog and cat must never be left alone unsupervised at all until you know that the dog won't chase, and the kitten is going to be sensible. Some dogs will never be safe around cats, but with perseverance most can be trained - I've had ex racing greyhounds living happily and safely with both cats and house rabbits, but again I wouldn't ever leave them together unsupervised, it is just too much of a risk.