Hi, Frank. Lets start with the easy part first. The depth that you see in furniture comes from the finish, not the stain. Dye stains will give you a little more depth than pigment stains, but I don't think that it is that noticeable. Many cabinet makers, particularly in mass production shops, spray on industrial grade finishes that look good and last a long time but don't have the depth. Furniture craftsmen use shellac, oil based varnish, or even wiping finishes but those don't have the protection that you need on your kitchen cabinets. So the best trade-off for both finish and protection is oil based polyurethane varnish. Since you will be using the white stain, which is heavily pigmented, you will not be able to get the depth that you see in wood without the heavy pigment. The good news it that the white pigment will cover a little of the dark pigment that was on before, but you will want to get almost all of the dark pigment off.
As for surface prep, with stripping and bleaching, you should be able to get out the dark stain. Try household bleach first and if that doesn't work, use oxalic acid. The bleach works on dye stains and the oxalic works on pigmented stains and there is no way to tell which will work other than trying them. One thing that you don't want to do is sand the stain out. Sure, it will take the stain out but you will also lose defination in the molded areas and cause problems in the corners. Certainly, you will have to lightly sand to knock down any raised grain before finishing but that's all you want to do. You might check some of the posts on www.citristrip.com for more on this. I'm sure that it will answer a lot of your questions or bring on a lot of new ones. Be sure to use the safety equipment recommended by the chemical manufacturer when using the stripper, bleach or any other chemical.
And you will find that the job is more pleasant if you take off the cabinet doors and drawers and work on these outside in the shade. That way you only have to do the face frames in the kitchen. And you can work on the doors and drawers horizontally which makes the work a lot easier.