Removing filters improves image quality, not adding them :)

UV filters should only be used for eminent dangers such as salt-water spray and flying sand. For protection against knocks and reduction of flare, a lens hood is far more effective.

UV filters as well as Skylight filters greatly increase the chance of flare which reduces image quality and contrast because it adds a flat (and relatively cheap) piece of glass in the optical path.

Skylight filters are pretty silly to use on a digital cameras since, as you suspect, the camera compensates for it when using TTL white-balance. On most cameras that means Auto and Custom white-balance. Preset and Kelvin white-balance will get the pink tint.

If your cameras has en external white-balance sensor (such as some Olympus and Nikon DSLRs), then it will not correct for the pink tint when it decides to use the external sensor and will correct when measuring TTL. You can sometimes disable the external sensor but these camera generally decides when to use the external one vs the internal sensor. To be sure to cancel the pink out, you need to use custom white-balance from an image taken with Skylight filter on.