Sounds like partially frozen kegs to me. Also the suggestion of finings could be on the money.
The ability to freeze is a combination of the freezing point of that fluid, the pressure and the temperature.
If you had a tray of water, it would freeze, if you carbonate that water in a bottle with pressure, it will take a colder temperature to freeze. So with your kegged beer, it is at a higher pressure and some of it is turning to ice. The water bit slowly turns into an ice slurry at the top of the keg, leaving a more concentrated beer at the bottom which you are dispensing. It is syrupy, not carbonated and will not taste like the beer you expected. It will taste maltier, sweeter, hoppier, more alcoholic and more bitter. These flavour components may be in balance and you have a barley wine. If you opened the keg, you would see ice slushy on the top.
Your beer bottles are under pressure and freeze at a lower temperature than plain water. Get that fridge a little cooler and they will freeze too. So it is probably a warning that your fridge is too cold. Use one of the digital thermometers with a flying lead to measure external temperatures, put the probe on the lead in the fridge near the bottom and leave the display sitting on top of the fridge.
Yes, this is a way of making a stronger brew, freeze your beer to a slurry, allow the strong beer to strain out and collect it. There is some discussion on HBD about it, including that it does not oxidise. In Australia, we are not allowed to concentrate our alcohol by icing a beverage.
