Has the rate of neutrino production in the universe varied over time? Are there more or less being created now against the early days?
Likely more Neurinos are produced billions of years ago than today because there were more very large stars in young galaxies. I'm not going to touch the first few seconds after the big bang. Someone else can if they want, but post star formation, it's a surprisingly simple problem to work out.
Neutrinos are produced primarily in nuclear reactions in stars. Large stars produce far more than smaller stars. A star with 10 solar masses, would, very roughly, produce about 10,000 more neutrinos. Stars that size are uncommon because they only live about 10-15 million years. Only about 1 in 800 stars are B-type or greater in terms of neutrino production and most of those are only a few solar masses, producing between maybe 50 and 500 times the number of neutrinos our sun produces. Very high neutrino producing stars are sufficiently rare that they don't throw off the numbers. (I'm ignoring supernovas, but I'll get to that).
If we estimate 100 billion stars in the milky way, the larger ones give off far more neutrinos than our sun and the smaller ones, far fewer. And our sun gives off 1.79 x 10^38 neutrinos per second. Doing some very bad estimating, all the stars in the Milky way together give off probably in the range of 1 x 10^49 neutrinos per second. Figuring our sun, being on the large size, gives off more neutrino to mass ratio than average, but not enormously more). Again, we're ignoring supernovas. You'll see why soon.
So, bad estimate, 1 x 10^49 per second from all the stars in the galaxy, or 3 x 10^56 per year. Or 1 x 10^58 in about 30 years. (Why do I care about 10^58 you ask?)
Because that's the number of neutrinos that the type II supernova observed in 1987 released. Source. So one type II nova produces the same number of neutrinos as all the stars in the milky way currently produce in about 30 years, give or take.
The milkyway has an estimated one billion neutron stars and about 100 million stellar mass black holes. Each of those can only form by type 2 supernovae. Given the age of the Milkyway, about 13 billion years, if we average it out, that's one type II supernovae every 12 years or so. The current rate is much slower, so the young milky way had significantly more type 2 supernovae than it currently averages, and, it would be consistant that there were significantly more large stars. Even with significantly fewer smaller and main sequence stars, the large stars and type II nova are the clear winners in neutrino production.
The current supernova rate (Type 1 and II) in the milky way is one every 50 years, so the rate had to be several times higher when the galaxy was young.
Peak Neutrino production would coincide with peak type II supernova explosions, which would only be 5-15 million years or so after peak very large star formation. I don't mention Type 1 supernova, because, while more common now, they (I think) produce far fewer neutrinos. Still a great deal, but well fewer than 10^58, though I had some difficulty finding a specific number.
It's very likely that young spiral galaxies like our milky way produce (very bad guess), maybe 10 times more neutrinos than they produce some 13 billion years after formation, so that would be the most likely answer. Once large star formation in galaxies was well underway, Neutrino production probably peaked, whether that was when the universe was a few hundred million years old or a couple billion, I don't know, but neutrino production likely peaked early with the formation of many large stars and began a steady gradual decline after the peak.