Short answers, pulled from the story.
Robert Hooke published Micrographia in 1665 after examining a thin slice of cork under a compound microscope. He described the empty chambers as cells because they reminded him of monks living in small rooms.
Robert Brown discovered the nucleus in 1831 while studying orchid flowers. This discovery preceded the formal cell theory which states all living things are made up of cells.
Cytopathology is the scientific branch that studies and diagnoses diseases on the cellular level. It generally uses samples of free cells or tissue fragments rather than whole tissues like histopathology does.
Scientists utilize fluorescence microscopy to label specific components of a cell with markers like GFP. A certain light wavelength excites these fluorescent markers so researchers can visualize them clearly.
Cell culture utilizes rapidly growing cells on media to produce large amounts of specific cell types efficiently. Large scale manufacturing of biological compounds such as vaccines and therapeutic proteins happens within these systems.