I completely agree with your preference for unusual names! It makes them more memorable and easier to look for at the same time.
Thinking about the possible origin of 'Eglantier', I had an idea it could have come from the French Republican calendar which replaced our usual Gregorian calendar from 1793 to 1805 and for 18 days in 1871 (Paris Commune revolution) and I was right. It replaced the name of the Saint associated with the 21th of Fructidor (~September 7th): http://en.wikipedia.org/wiki/French_Republican_Calendar
Either his parents were resolute Republicans or more likely he inherited the name from a family member (grandfather, uncle...) or perhaps even his godfather. You'll have to check his siblings names to see if there is a trend there or if he's the exception...
About the handwriting, I've found that deciphering records written in your mother tongue is always easier than in other languages, no matter how well you speak/read them. I think it's a question of mental reflex, your mind automatically trying to find the match in your usual language before trying something else.
Finally, about the mention of the mayor and midwife names in the certificate: the name of the one writing an official record is always specified, I think it must be a legal obligation because it's still the same nowadays, though it's generally done at the end instead of the beginning (but each region has its quirks). The midwife is less common but is often the one to declare the birth if the father is absent (like in this case) or unknown (single mother). It makes sense though because she was the main witness of the birth process, so unless a family member (or sometimes neighbour) is sent to notify the authorities, she's the natural informant. Note than in France you must register the birth of a child within 3 days at the Town Hall (and it's mandatory), after that you'd have to petition the Court to have it recorded. The address, it gets more complete as time goes by, but indeed it never has been a priority, and you should consider yourself lucky that the street name is even indicated...
Good search! :)