I can't speak to the prevalence of wigs/toupees at the time, but men did not tend to shave their heads to hide the pattern balding the way they do today. Growing your hair should be fine.

The most fashionable facial hair in the 1880s was a dapper curled mustache, as seen here and here. However, beards were certainly seen: they could be trimmed close and with a slight perkiness for fashionability. (Fuller version.) Less-manicured beards had been more popular in the 1860s and 1870s, so by this time they were more likely to be associated with old men or artistic/philosopher types. (Less-manicured mustaches and things like mutton chops were also still seen, but more dignified than fashionable.)

I can't speak for France, but in Britain the style at the time was short well groomed hair with a parting coupled with a big bushy full beard, or mutton chops. I can't gamine French style was all that different to British.