How can I remove a stud to fit a range hood vent?

Options that avoid moving the range or knocking open the wall:

1. Go for an unvented hood
2. If there's no top cabinets you could get a ducted hood and run the duct inside the kitchen across the wall then outside where possible. The choice of hood, size of the duct, and number of bends all have to be to code and planned by someone with more knowledge than me but I hope this can give you some direction if you like the idea. With some thought this could be made to be not too ugly, e.g. an industrial-look duct, or build a plenum around it.

Give us a picture of what it looks like now.

Assuming the current hood doesn't allow for the duct configuration required to work with the existing situation, Aside from possibly creative (unsightly) ductwork with a range hood UL listed for multiple bends in close proximity to the connection along with likely losing some cabinets if room for the ductwork is needed and ducting through the roof, there are no other safe/code approved solutions with the current stove type. You may need to get a lawyer, start with a demand letter to the contractor. As @HABO notes, you might be able to replace your range with a downdraft unit, but there might be just a many issues below as above...

If it's an electric range in a jurisdiction governed by out of date regulations that don't prescribe any exhaust requirements and a more "libertarian" minded inspector, a certain type of homeowner might just cut through the stud as is as an unpermitted renovation after the fact, careful to move the wire out of the way, as long as the existing permit can be closed without a vent in place, and as long as such a renovation doesn't violate any of the regulations in your jurisdiction including the need for such work to be permitted. A certain type of homeowner might be willing to take the risk that further modifying a single stud that is already rendered structurally irrelevant by a pipe running through it above is an acceptable level of risk, and doing so without the intention of transferring its load "sistering", and that type of homeowner might look at such a stud as really only serving the purpose at this point of fastening media for the siding because if the little section of wood left by the penetraction near the top plate really is the Las thing holding up your house you've got bigger problems.

There are so many options that this question is a bit unclear.

#1 - Don't vent. You have a window. Already meets code.

#2 - Run a bulkhead go out where you want.

#3 - Buy a kit for your vent to make the top of your vent unit run straight up (bigger). Use that space to figure it out.

Your question isn't about how to vent. It is how to vent and it looks nice. There are tons of custom tops for vents that hide this stuff or you can drywall that part. I see almost a 5" gap that you could go straight out. Sorry this just has so many options not really sure what you are looking for.

Also usually 6" is required for a round duct by most inspectors. But given my point #1 you really have no requirement so you can install a 5" duct. Or you can get ducts in "code" sizes like the 3.5x10 - just an example I have used I am sure there are a multitude of sizes... There are lots of options. You do not need an 8" duct unless you are in a commercial kitchen.

Update: Given the last round of pictures your question is much different. It was, what can I do... now it is holy crap this won't work because I have a finished kitchen. So given how nice your kitchen is finished and the fact that you would be INSANE to disassemble this at this point there is only one solution.

1. I have a local metalworks shop build you a custom vent adapter that will fit this and go out the 5" opening. They will just need to fit it to the 8" pipe and honestly it shouldn't be a big deal to do. (labeled #1 below)

2. The only other option is that they trim out the top - create a larger metal box/soffit and go out label #2 below. This could look great or be an eyesore. This will probably be costly to do if it looks nice.

3. Don't vent, you have windows right there.

4. Buy a different venting system that allows a smaller output or one that has a larger top so you can go out a bigger area. The vent should have been hooked up and ran before finishing the wall. Period. You cannot be picky about getting exactly what you want now unless you are willing to really open up that wall - might mean losing all of the tile - or at least a big big chunk of it.