Legal obligations for certain kitchen appliances

no

I don't know about 'legal', but I doubt it very much as one can rent 'unfurnished' which means what it says. Of course, if it had to be a built in one then its absence would be odd. You could, however, negotiate. Some landlords may well have a spare, particularly if they have other properties. It's not a big ask to seal the deal.

No - none whatsoever.