Yes, the whole bathroom floor becomes najis in the said scenario with the existence of dampness, and whatever touches it with dampness becomes najis, provided that this was the only cleaning method used. If the floor is cleaned through other ways as well in which the taharah of the floor is achieved, such as with running tap water, such that one is no longer certain of the najasah of the floor when it comes in contact with the foot or the clothes, then one can consider the foot as tahir. This applies to any situation where najasah is doubted after being sure that taharah took place.
Tayammum should be performed in the following method: One should first strike the inside of his hands upon the earth once. He should wipe, with both of his hands together, his forehead, and the two sides of his face from the place where the hair grows to the bone of the eyebrows. He should wipe the back of the right hand from the wrist to the fingertips, with the inside of the left hand. Then he similarly wipes the back of the left hand with the inside of the right hand. It is best that he strikes the earth for a second time and then repeats the wiping of the two hands.
This does not make everything najis, since one is not aware that they had been in contact with najis things with dampness, or one is sure that a particular thing became tahir and najis without knowing which one was first. In such cases it is considered to be tahir.