No. A solid, clean and flat surface is required for installation of ceramic tile.
The best answer would be to remove the flooring and inspect the sub-flooring for any rotten wood or any issues. However, If you wish to skip that step you can lay your cement board over top of the vinyl then your tile. (If your question was to just install tile over the vinyl without cement board, then no you cannot. You need the proper base ie. cement board, ditra or anything other comparable product).
In most cases the answer is no. If the vinyl floor in question is a non-cushion-back vinyl or a linoleum, some manufacturers have adhesives that can bond to the floor with the proper preparation. In addition, 1/4" underlayment most commonly used under vinyl isn’t supportive of tile installations.If you’re not sure about the type of floor you have, or if you don’t know what the flooring underlayment is, you should plan on removing the vinyl and underlayment.
A supportive and properly prepared subfloor is the most important factor in the longevity of your tile floor.
Most definitely not, the vinyl floor will have to be removed and installation of a backer board (such as wonderboard) should be bonded to the subfloor prior to tile installation.