"Should we have the hood about 6-12 inches higher than the cabinet top or keep it even with the cabinets?"

The dimension you need to focus on is the distance from the cooktop to the bottom of the hood. That distance should be between 30"-36". The "cabinet top" would only matter with certain hood designs as to the height the hood is placed. You hang the hood so that the bottom of the hood is between 30-36" from the cooktop.

Can you provide more information or a photo of the hood location? I cannot tell if you will have a chimney above the hood, or have the hood hung under cabinets, or have a soffit, or what arrangement you are looking for.

Here are some photos showing various ways a hood is hung from the ceiling or from under cabinets. Many different widths and lenghts of chimney extensions - if that is what you are talking about. Again, more description - or perhaps provide the hood names and model numbers that you said you are looking at.

Height above the cooktop is what is critical. That will be a function of hood power as well as depth/width. A bigger, deeper, more powerful hood can be placed higer above the cooking surface and still work. Once you know where the bottom goes you trim out the top to look good.

I got a laugh from the OP’s original comment: we have 14 inch high ceilings…