[Verizon Galaxy Note 3 Q&A, Help & Troubleshooting] bftb0:
There are probably several things worth checking.

1) Is this something that happens only immediately after the ROM install, or persists indefinitely?

The best unassisted (meaning no aGPS) GPS module will take 60-180 seconds to acquire a first fix from a "cold start" condition, depending on satellite constellation configuration at that moment. It's not really possible to improve on that without differential GPS.

So, that "10 second fix" can really only happen in one way: by having a precise time estimate and also a good previous fix.

Before the advent of aGPS (Assisted GPS) in phones, handheld or marine GPS units would engage in a kind of trickery: they would always save the last fix acquired when shutting down, and then use that as an AP (Assumed Position) the next time they were powered up. A good AP guess can dramatically lower the time to a first fix if it happens to be close to the actual location.

And it just so happens that a lot of people experience that exact usage case: they shut off their GPS in the evening, sleep overnight, wake up and power up the GPS... just a few feet feet away from their last fix. Or consider a boat owner - they turn their GPS unit off when they return to port, and the next time they depart days or weeks later, they turn it back on right at the location of the previous fix (the harbor). Voila! A rapid 10-second fix!

Now enter aGPS; the cell network can give the phone very precise time and also a position to within a mile or so. Now it doesn't matter if you shut the device off, carry it cross country, and power it back on - if it is able to talk to the local carrier, it will have the precise time and a crude but decent estimate of position in a few milliseconds. That AP (Assumed Position) collapses the time required by the GPS to produce a first fix, and Voila! A ten-second fix!

(Remember that in the US, ALL cell phones - even dumb flip phones - have a GPS chip in them to comply with national 911 regulations; the Man always knows precisely where you are if you are in cell range... even when you "turn off your GPS")

So, the "10 second first fix" is a little bit illusory - it is not something the GPS unit is capable of by itself without assistance. Turn off your phone, fly to a different part of the globe, travel to a spot with no cell service, and turn it back on again... and it will take even longer than a cold start to get a first fix... because it starts up with a horrendously wrong "first guess" AP.

The first (Verizon) Android phone I had, I remember other owners claiming that GPS would not work at all when they were in Puerto Rico (roaming on another carrier)... and made the claim that the GPS chip was so heavily dependent on aGPS, that it simply could not ever get a fix without it. (The presumption was that the local carrier did not have any aGPS servers.) Later, I had the experience of using the same phone in wilderness conditions where I would keep the phone either off or in airplane mode to (save juice), and occasionally fire up the GPS to record a special location or two. Many times - whilst outdoors with a horizon-to-horizon view of the sky (with 6 or 7 birds up), that phone would take over a minute to lock. But it would eventually do so.

It is still possible that those folks were correct about the Puerto Rico thing - in my case my prior aGPS fix might have been 60 miles away, "in town"... but those other folks may have traveled closer to 1000 miles with their phones off. So my "last fix AP" guess was crude, but far less crude than theirs. (Either that or they were just impatient, and never waited the minute or two required for a "real" unassisted GPS fix in a brand-new location)

So I mention the above because a lot of stuff has to hang together to produce that "10 second fix"; an effort to decipher where the variability is coming from has to look at a large number of possibilities.

I don't know if this device (VSGN3) uses proprietary firmware device drivers ("blobs") for the GPS end of things - but if that were the case and a ROM was ported from a different carrier and the firmware blobs were not replaced correctly, that could account for something?

As for fix accuracy, remember that the GPS "Accuracy" is not the actual positional error - if you "knew what the error was", you could simply eliminate it by subtraction. At best it is a stochastic estimator based on signal strength estimates, positional diversity of satellite constellation, individual path delay estimates, etc.

Meaning: it is an estimator of system noise. You have to take a ton of measurements of noise to say anything meaningful about it, and a whole lot more than that to compare two noise distributions with any resolution.

So I would be careful about coming to the conclusion that the accuracy is worse - it is possible that you just happened to look when the constellation was poor.