A person who lays tile can be called a tile fitter or tile layer. Some people call it a tile mason, too.
"Lays" is correct for the third person singular present indicative of the transitive verb "lay", as in "Lay the tile quickly so that we can leave." A single third person who obeys this request lays the tile.
A tile installer or tile setter.
A tile installer or tile setter.
masonry
A bricklayer or bricky
The small tile parts of a mosaic is called tesserae
A shingle...
A person who lays wooden floors is typically called a flooring installer or a hardwood floor installer. They specialize in the installation, repair, and maintenance of wooden flooring. In some contexts, they may also be referred to as a carpenter, particularly if they work with various types of woodwork.
Calli (call-ee)
Wherever he lays his hat.
This really depends on where you are located and how much you are willing to spend on the repair. It might just be cheaper to buy a new tile cleaning machine. I'd call around to local repair shops and the manufacturer.
The small tile parts of a mosaic is called tesserae