Were it me, i would rip it all out because you have the area in front
of the dishwasher to deal with and it's starting to get pretty thick.
I had a house once that had 5-6 layers of vinyl and a couple layers of
plywood/luann and it made the floor joist bow down in the middle. So
*I* would remove it.

The asbestos deal is something to look out for but may be removable
without too much fuss, though ymmv. I have what seems like 12 million
sq feet of the stuff (see previous post - it's 9x9 inches and looks
like that and if you break one, you can see the fibers) and it comes up
very easily cause it's been there for 60 years.

As for what to put as the flooring - I am not on the laminate
bandwagon, I've seen friends that have it and it is affected by
moisture and heat a little bit. If you never think you'll drop a whole
2-litre bottle of wine like my roomate did, then you're ok. I just
worry about that too much I guess. Kitchens = ceramic in my book.