Does ritual impurity transfer if a six-year-old child enters a bathroom whose floor is ritually impure with urine, and then the child touches objects like doors and devices, or touches people with his contaminated hand, and is it obligatory to wash what the child touched or to wash the person he touched?

If you are afflicted with waswasah (obsessive whispering/doubts), you must turn away from it. The default state of things is purity, and this includes the bathroom floor, unless you are certain of its impurity. If the foot becomes soiled by a wet impure spot, it becomes purified by passing over a pure spot afterwards. Mere doubt does not render things impure, and if a child's foot becomes impure and then dries, the impurity does not transfer merely by touching it with the hand. The default rule when in doubt about the transfer of impurity is that it has not transferred.