Is Sudoku Always Solvable?

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Maybe it can help you to exclude some configurations. Yes, it will always have a solution as long as you start with a good solution and then make legal moves to randomize the tiles. BUT if you, for example, pop two pieces out and switch them, then you can get an insolvable puzzle. 😉 As you discovered.

Is it possible to make a puzzle that always has a solution?

Yes, it will always have a solution as long as you start with a good solution and then make legal moves to randomize the tiles. BUT if you, for example, pop two pieces out and switch them, then you can get an insolvable puzzle. 😉 As you discovered.

Is it possible for a random shuffle to result in no solution?

If it is possible that a random shuffle could result in no-solution, then I’d have to figure out how to verify that there exists a solution, which is a completely different beast. Martin Gardner had a very good writeup on the 14-15 puzzle in one of his Mathematical Games books. Sam Loyd invented the puzzle.