If your flooring is attached, like hardwood or tile, you have the option. If it's a floating floor, many of them are not allowed under cabinets (of course you can do what you want but voids warranty). I've seen one LVP that is allowed under cabinets if it's glued down. So if you have flooring picked out that might make your decision for you.

IMO as an installer and remodeler, it's generally a waste to put kitchen flooring under cabinets. Generally by the time you decide to change your cabinet layout, the flooring will be due for an update too. It's very seldom (if ever?) that I've seen someone want to move kitchen cabinets and keep the original flooring.

On the other hand, in higher-wear rooms like bathroom, laundry, mud room etc, I recommend flooring before cabinets because the odds of replacing those cabinets sooner are much higher, and the amount of extra flooring is smaller.

If you REALLY need to find studs and nothing else works, do this: pull off the baseboard trim and start drilling holes 2" off the floor- high enough to miss the bottom plate but low enough to be covered by the baseboard. Once you hit one, you should be able to measure out the rest.

I would simply remove the old handle, bore the door and install a standard handle.

That's baseboard not skirting, and you should be caulking not grouting.

Taping & caulking a straight line is kind of a skill learned by experience. I can do it after years of practice, but I'm not sure how to tell you how to do it.

An alternative is, you could match the caulk color to the trim (white) and tape the floor.