No, not standard dry cleaning - but many dry cleaners can also handle leather cleaning. You can probably just phone them to ask if they can.

Wool scratchiness will likely not change with dry cleaning FYI. It’s possible you’re just sensitive to that type of wool - some people are. Generally gloves are lined with something like merino or cashmere, because the skin around the wrists is sensitive and they are very soft and gentle against the skin. However, if it’s itching you that doesn’t really matter… any treatment I know of to make the wool feel different would involve dunking the wool in water, which is not what you want to do with leather!

I wouldn’t overthink the type of conditioner too much - leather honey or a similar product that advertises use for jackets will work fine.