I ran mine tuned for many miles and a good number of trips down the quarter mile with the stock thermostat. I did finally put a 160 in this year. However, Anthony at LME retained the stock fan settings on the tune that I run. Many others with the 160 have had the fans set so they don't run so much. I also have a tune where the fans turn on very early and run constantly. I have used that tune at the track a couple of times. Once fully warmed up, the temps are the same at around 185 on my MKS regardless of which thermostat is installed.
The 160 will take a bit longer to get to 185 and it will cool off a bit more quickly when you are coasting under no throttle. WOT runs through the quarter mile will raise the temps with both. Many have noted the 160 will not reach as high a temp as the stock when under WOT. With mine, I see around 210 or so regardless of which thermostat is in the car. The 160 does cool off more quickly, though, when slowing down at the end of a run. I have a 2010 MKS and a 2014 Taurus. The MKS does have a larger grill opening and tends to run a bit cooler than the Taurus when moving and air is flowing through the radiator.
As for burping, I raised the passenger side of the car and filled the bottle with the same amount that I lost when I changed the thermostat. I ran the car until it was warmed up, held it at about 2500 RPM for a few seconds, let it idle a few seconds and repeated that rev/idle process about 3 times. It sucked the new coolant into the engine and once the car cooled down, I made sure the coolant was at the max level. I checked it a few times after that and it never moved. Burping was pretty much a non-issue.