In what years were fountain pens used?

That depends a great deal on your definition of fountain pen. If you include quills (basically feathers with sharpened tips) then it was some folks before 900 AD. If you include dip pens, then some folks in the 1700s.

Otherwise, The first known description of a pen that could hold its own ink is of a french design by M. Bion for the Plume sans fin

• feather without end in 1723.

1819- The Penographic was patented. Inkflow was managed by pulling on a valve behind the nib.

1832 John S. Parker patented a piston pen.

1884 Lewis Waterman had just lost a sale to a wealthy client because his pen spilled ink all over the document. He decided to improve on the design and patented his own design for an eyedropper filled pen that would not leak.

1900 or so Conklin started selling a pen with an internal bladder that could be squeezed by a protruding metal crescent. Other pen makers tried several ways to emulate that design without infringing on the patent. This lead to match fillers and coin fillers to push the metal bars against the sac. Sheaffer introduced a lever filler that outsold everything else. Other pen makers tried various designs until the 1960s when cartridge filled pens became the norm.

Many designs came and went over time, and some of the older designs can still be found today. So, who invented it? Maybe Monsieur Bion, but Waterman got more publicity