Creating a vacuum above carbonated drinks causes the CO2 to outgas faster--simply because there is no CO2 above the drink to diffuse back into the liquid. In physical terms this means there is no vapor pressure of CO2 above the liquid, so net movement of CO2 is from the drink to the space above it. If you leave a closed carbonated drink bottle long enough, the partial pressure of CO2 in the drink and in the space above the drink are the same--rate of gas that escapes the drink is equal to the rate that dissolves back into it--equilibrium.
Note that it is not solely about gas pressure but partial pressure of the gas you are interested in. Pressurising your champagne bottle with pure air to above atmospheric and sealing it wont extend the bubblyness. You need to pressurise with CO2 gas. For soft drinks you need 2 bar CO2 (in a typical coke bottle left alone for awhile, the space above the drink is almost pure CO2 and is at 2bar).
Your question about PET bottles is a good one. As mentioned, without a CO2 pump you cant extend the life of your sodas. Squeezing a half full bottle and sealing it removes the volume of available space above the drink, so when CO2 inevitably escapes the liquid to form equilibrium vapor pressure above it, less CO2 is required for this to happen--if the bottle doesnt expand back. The problem is that the bottle has structurual integrity and is designed to spring back. This creates low pressure in the bottle, causing CO2 to outgas faster than if you hadn't done all this in the first place. The upside is that equilibrium is the same--you lose the same amount of CO2 from the drink as if you had not done this. To really save gas, when you squeeze the bottle and seal it, you have to find a way to keep the bottle from springing back. Or just store unfinished soda in bottles where there is very little space for CO2 to outgas into.