What is the single most valuable move in FreeCell?

Emptying a column. It doubles your move capacity, and a second empty column doubles it again. Almost nothing else changes the shape of the game that much.

Should I fill an empty column?

Only for something concrete. An empty column is worth more as capacity than as somewhere to put a King — unless putting that King there unblocks a real sequence.

Why did I lose when FreeCell is almost always winnable?

Usually cells. Filling all four leaves you with single-card moves, and by the time that is obvious the position is already lost. Watch the move-capacity number in the bottom bar — when it reaches one, you are in trouble.

Is there a solver that always wins?

Yes, FreeCell is thoroughly solved and solvers win essentially every deal. That is exactly why a human loss is a planning error rather than bad luck.

FreeCell Strategy — How to Win Almost Every Game

FreeCell StrategyFreeCell is nearly always winnable, so strategy is everything: plan before moving, empty a column early, keep cells free, and dig out the low cards.