Do proteins move inside the cell and how fast do they move or move? Please support your answer with articles or books.
Proteins move in cells or get transported in the cell. You can go to NCBI (https://www.ncbi.nlm.nih.gov/), and start searching for protein dynamics within cells. Also there is a technique called FRAP (https://en.wikipedia.org/wiki/Fluorescence_recovery_after_photobleaching) which is used to measure kinetics and dynamics.
Also, this differs probably a lot between the cellular system you are asking for. You can imagine that some cells have a more active transport system because stretched out (neurons) vs others with maybe less active transport systems, because of a more static nature.