Yep, definately fleas, the fleas are dark brown or black and a bit shiny, they run very quickly through the fur and as you have already found the eggs are white and are laid in the bedding or wherever the cat sleeps. You need to be careful with young kittens, if they become overun with fleas they can become anaemic and very ill or worse, good news is you can get flea remedies for kittens although as yours is 10 weeks old she will probably be ok with adult remedy. Don't bother with any of the shop bought stuff you will be wasting your money, it don't work. Best wait till you go to the vet and buy from them, they will advise you on the best one, it will be more expensive than shop bought but cheaper in the long run cause it works.
I buy the Front line spot on, and also the household flea spray from the vets for round carpets and furnishings, the front line comes in a pack of three doses and costs around about £11 or £12, can't remember exactly, I have four older cats so have to buy two packs but only need to use it every 6 months to a year and that keeps them at bay.
Also wash your kitties bedding as the eggs can stay there for months before they hatch out.
Good luck with your new kitty, she sounds happy, what's her name?
You will get lots of good advice on here, look forward to seeing some photo's