There isn't only one proper way. If it's convenient to do so, extend the vent straight upward. If there is attic and roof immediately above then you could make a new roof penetration for the vent and detail with a pipe jack flashing in the usual way. If there's another story of the building above, you could get into that ceiling/floor space and take the vent ...
I do not think that it is technically a "washer".
It is more for rinsing a glass, by itself it is not enough to wash the glass.
These are used in bars for draft beer. If you rinse a pint/beer glass with cold water just before you fill it with draft beer it will prevent the the beer for foaming to much which will result in to big of a head on the beer.
It looks like my glass/ bottle washer. Is there a knob that turns the hot water on? Mine only had hot water ran the sink until hot set the glass on the wash push the button it would go for 20~ seconds great for rinsing home brew bottles out, those I had to hold the bottle in place because I lost the holder.
Might be a "washing machine adapter" or "dishwasher adapter" - same thing, really - a quick-connect fitting to allow use of an appliance via hose at full-bore flow. Not the same as the one I use for my bottlewasher, but I'm sure there are multiple incompatible designs out there. Anyway, you can unscrew it and have the normal threads available.