I also can’t provide a definite answer, but I suspect the answer to both questions is “not really”.
The Julian Day system’s great strength is that makes it really easy to calculate the time interval between two dates/times, which is difficult in most calendars. But unless they’re assembling historical chronologies, or doing astronomy over long periods, how many people actually do much date arithmetic?
The system’s weakness is that its epoch was nearly 7000 years ago, so Julian dates are big numbers. In the past, scribes got lots of practice at writing out big numbers that start 24…. But nowadays it leads to loss of precision in computer programs. Computers typically do calculations to 16 significant figure precision. If you waste some of those figures on 245… you have less precision on the digits that come later. A Julian Date stored in a standard-precision variable in a computer program is only stored accurate to 20 microsecond.
Many of the applications today that involve time arithmetic involve computer operating systems where a lot can happen in a microsecond. They usually use Unix times, stored as the numbers of seconds since 1 Jan 1970. That gives you 0.1 microsecond precision. I think pretty much everyone other than astronomers use Unix times rather than Julian Dates nowadays.
So in short, it’s not clear to me many people outside astronomy would have had much use for Scaliger’s system. And it’s not clear his decision to use an epoch in 4713 BC was ever a useful one. Is it fair to suggest today, even astronomers only use the system out of tradition, rather than because it’s good?