Put the place name in amenities ?

If the official name of the train station is "Worcester station" the name tag should reflect that. However if the official name is "Shrub hill" then you shouldn't add either "worchester" nor "station" to the name.

Either way the fact that it is a train station should be expressed through the tag railway=station and the fact that it is in Worcester can be inferred from the boundary way (or relation) of Worcester.

As for the bakery or the pub: If the name is "Worcester pub" then use that in the name tag. If it is "The prancing pony" then use that.

I'm not sure what the naming scheme for police stations in your country is but around here police station might have a name like "Worcester Police Station".

However there might be a second police station in town with the name "Midlands Borough Police Office" which not a 'regular' police station but the superintendence of all the police force in the Midlands.

If Worcester is big enough it might have more then one police station, say "East Worcester Police Station" and "West Worcester Police Station".

Perhaps, and I'm really out on a stretch there, there might be a third small police station in town called "Friar Constabulary" for historic reason. That station should be tagged with amenity=police, name=Friar Constabulary. The fact that it is in Worcester can be infered from its position after all.

As a summary: Use the official name in the name tag without adding words for "this is in Worchester" or "this is a pub/police station/whatever".

I would say Yes, if that is that is how the amenity is officially known or is signed on-the-ground. A city may have more than one train station, post office or police station, so the distinction can be important.