بسم الله الرحمن الرحيم
(1) It is the etiquette of ghusl that one performs it wearing lungi etc. However one is allowed to take a bath naked in such a place where no other person can watch.
(2) It is allowable and proved from the Holy Prophet Muhammad, may peace and blessings of Allah be upon him.
(3) If the baby boy or girl are too small i.e. of four years or less then it is allowable to see and touch them while giving them a bath. But when they turn older, then both the private parts are satr till they have sexual feelings and shall increase with their growth. However the boy of ten and the girl of seven years of age shall be considered mature and no parts of the body below the navel up to knees should be touched without a cover as well as the girl must not be touched. And it shall be completely necessary to avoid touching the girl after she turns nine years of age. It is agreed upon by all that a girl starts having sexual feelings at the age of nine. There is fear of Hurmat-e-Musaharat if the girl of nine years of age is given bath naked with hands.
(4) The method of ghusl for male and female is same. First, wash both the hands up to the wrists, wash the private parts whether they are smeared with impurity or not. Further, wash the body part where impurity is smeared. Then make complete wudhu. However if one is taking bath in such a place where water lingers near the feet then one should not wash the feet. One should wash the feet moving from there after completing the ghusl. Then one should pour water over whole body in a way that water reaches up to the whole body. In case the hair of head are not tied then one should wash all the hair as well as let water reach their roots. In case they are tied then it is not mandatory to wash full head rather it shall be required to let water reach the root of all the hair.
(5) It shall be lawful to take a bath in that bathroom after pouring water in such a way that the impurity is flowed out of the bathroom. However one should avoid taking a bath in such place as it shall create waswasah.
(6) Yes! It is impure. It is mandatory to wash the place where its urine is smeared. However it is not necessary to wash the urine of newborn baby with much efforts.
(7) Consult any expert Hakim for the same.
(8) The method of cleaning such chuddar or mattress which cannot be squeezed is that it is washed thrice and left for as long as the drops are stopped. It is not necessary to dry it each time completely. However you may use machine to dry water from such floor. The mattress shall not turn pure in the way you mentioned. However it is better that you spread some chuddar on the mattress before having intercourse so that the effect of impurity does not reach to the mattress.
(9) It is not necessary. However if there is fear of smearing with impurity then it shall be necessary to go there wearing a slipper.
Allah knows Best!
Darul Uloom Deoband