The general range for the height of a sink is about 32" to 36" give or take.

Vessel bowls vary from 4" to some I've seen that were over 12" tall, so you'd have to choose your bowl then size your vanity to get to your desired final height.

An open or closed vanity may depend on the look that you're after. Clean slick contemporary? Something else? The choice really is yours, and the practical nature of available storage does come in to play.

As far as chopping down an existing vanity, it depends on how much need to be chopped as well as the style and construction of the vanity.

While it's easiest to chop the feet off or eliminate the tow kick on a cabinet, having it simply sit on the floor may look a bit awkward.

While more difficult, it often look better to take material off the top, then add a new cabinet top that will compliment the sink. Then doors need to be changed as well.

Paint grade you can dicker with, as the primer and paint can hide some scars. Stain grade, it might be easier to just get a new cabinet.

Mongo

We didn't change the cabinet when we added our vessel to the powder room. The cabinet is 31" and the vessel adds about another 7" and the faucet is quite tall. It works fine for us.
I had the laminate counter top removed and put in a cherry top to match the cabinets.

Our cabinets are 34.5" plus the granite. Seems tall, but it's just fine for our average heights. We can lean right to the edge of the sinks and there is very little splashing.

Another option to consider is to semi-sink it into the countertop. This reduces the height of the vessel. This works well, and looks real nice with a granite top.

I just remodeled both of my bathrooms and did vessels in both. I built custom setups for both...the master got a floating wall mounted shelf and the guest got a more tradional vanity with doors..but it started life as a drop in height vanity. I cut off about 4" off the top to get the over all height just right. The doors are perfectly intact...then made a custom counter top from MDF and sat a nice round white porcelain vessel topped off with a glass waterfall faucet.
Both projects are easy DIY. I have photos over at my site.
Cheers
Karl

Here is a link that might be useful: Bathroom Daddy

Height was a big concern for me - I didn't want to feel like I was reaching up and in to the sink. I selected (ordered, but not yet installed) a 29" high vanity. The vanity top and the sink would be added to that height, putting the final height the same as most undermount sinks.