You can get a basket and layer it with warm blankets, and/or towels. Fleece blankets are ideal.
Any box, or tub works well as the base. The main purpose of the bed should be to keep the kittens warm, since they have trouble regulating their body temperature so young.
Yes, the kitten fell off the bed.
Yes, the kitten is tucked in bed and sleeping peacefully.
yeah if its safe
to get warm and settled in to sleep
If the kitten is young, it may be because he/she isn't fully litter trained yet. If the cat is over 6 months old, it may be habit - get the dog a new bed that doesn't smell of cat pee and see if that fixes the problem. If the kitten still urinates in the dog's bed, you may want to talk to your veterinarian about training options and double checking for urinary tract infections.
To prepare for your kitten's first day home, make sure to have a cozy bed, food and water bowls, litter box, toys, and a scratching post ready. Create a safe space for your kitten to explore and adjust to their new environment. Schedule a vet visit for a health check-up and vaccinations. Spend time bonding with your kitten and providing love and attention.
become best friends with your cat and it will want to be around you more and eventually get in your bed. The top answer would work but I suggest not to sleep with your kitten until it is a bit older because you could move in the night and squash it. Also the kitten could fall out of bed and hurt itself. Most cats tend to sleep on their owners beds anyway. 
Kitten saliva may carry bacteria or parasites such as Pasteurella or Bartonella, posing small health risks especially to immunocompromised people. While infections are rare, it’s wise to avoid lip contact and practice proper hygiene.
There are some diseases that a sick kitten can pass on to make a child sick.
Probably about 1 year . You can make one unless you get your mama cat fixed then you cant make a kitten.
Your kitten is unfamiliar with his or her new surroundings and will come out when he/she feels comfortable ; just be patient .
Yes. A sick kitten can make other cats sick.