I think the fold back door are a great idea for a washer and dryer that has to be in the kitchen. I think the average swing door would be a pain. Yea, you might have some vibration from the washer, but as long as the washer doesn't hit the sides or front of the cabinet it shouldn't be a problem. I wouldn't tie in the floor that the washer/dryer is on into the cabinet frame.

Here is my setup. The vibration isn't an issue at least with the Electrolux models I have which also require zero clearances around the sides and back. The only issue I have with the arrangement is getting them in and out of such a tight space is a bit challenging including getting the vent attached to the dryer. I eventually found a magnet attachment that helped a lot.

To make the w/d flush with the others they were recessed into the wall about 6" which wasn't a big deal given the construction of the addition.

I liked the idea of the pocket doors but ultimately skipped them. The units are really too wide for a single door and the hardware for 4 doors ends up taking a fair amount of space. Cabinet maker talked me out of it. No regrets there.

A non-permanent countertop sitting on top of washer dryer.

Lots of information on below link of people who have done different options, explaining pros & cons

Counter top on front-load washer/dryer

[https://www.houzz.com/discussions/counter-top-on-front-load-washer-dryer-dsvw-vd~769854[(https://www.houzz.com/discussions/counter-top-on-front-load-washer-dryer-dsvw-vd~769854)

#1 source for in home floods is the washing machine. #2 source of home fires is the dryer. If you don’t build to handle those right now emergencies, then you are wasting time and effort. It is a utility area, not a formal drawing room.

Your question was "...if it is a good idea to put our washer and dryer in kitchen cabinets?" and I think Sophie gave you her answer to that question, looking at potential issues that are at play.

Have you seen this thread? I didn't read all the above replies and maybe it was posted. I adore this setup in a remodeled bungelow by @caligirl5:

[caligirl5 bungalow kitchen reveal[(https://www.houzz.com/discussions/bungalow-kitchen-reveal-dsvw-vd~4882448?n=57)

I think what Sophie means since you have no where else to put the washer and dryer to take in to account the safety issues. Make sure you have a good pan under the washer with a drain and that you have the best hoses for the water supply. It also would be good to have a water valve in easy reach to shut off the water when the machine is not in use. Water can spray all over the kitchen if the hose fails. The pan will not be much help in that case.

Also besides cleaning out the dryer lint every time you use the dryer, you should be able to check the dryer vent periodically for built up lint.

What's three best way to get the dryer flush with the wall?

https://dryerbox.com

or equivalent

You should also have a recessed water valve where it's easily accessible, and not plastic. I used to have one similar to this, which made turning the water on for a wash and off once done a very simple matter:

https://www.homedepot.com/p/1-2-in-Bronze-Sweat-x-MHT-Washing-Machine-Shutoff-Valve-1-2-175C/205893456

There are a number of those available from different manufacturers.

ETA But personally I would prefer not to have a washer/dryer in the kitchen unless there's no other place for them to go.