Yes! Fossils are made when something living dies and is buried in the ground -- it can takes thousands of years for it to become a fossil. Then when we dig them up, although the original living thing is gone, we have a mineralized copy of it (which is the fossil).

For example, if a clam dies and is buried in the sand, over a thousands of years it might become a fossil. As water flows through the rocks years later, the shell will dissolve, but other minerals in the water might replace the shell. This happens so slowly that the fossil is formed to look exactly like the original clam shell.

Yes, more or less. Say you have a piece of wood with its cellulose structure. Solutions that have dissolved silica can saturate the wood and then as the water evaporates leave behind silica. The form of it will be governed by the original cellulose woody structure. The process is called permineralization ( read more here )

Yes, that is exactly how petrified fossils form: the minerals precipitate out forming a replica of the organism's body.