Take a look at the Yellowstone caldera system. It's a series of overlapping calderas with a great deal of geothermal activity.
From your diagram, I'm not sure what the vertical extent is, but what you've proposed is generally plausible. The river would probably not be a single channel, but rather a set of smaller channels rising through fractures in the rock. How large the total river might be is, of course, a matter of preference.
Depending on the particulars of the mountain, you might also have a glacier at the top whose melt supplies a large portion of the total water flow.
With regards to the palace, it is probably easier to work with hot water than with steam. Steam implies pressure and volcanic gasses tend to be pretty corrosive, which would lead to lots of nasty pipe bursts.
I gather from the descriptions that one of the important parts here is the presence of lots of steam and fumes that will wreath the architecture. Depending on how far away from the volcano the palace resides, you might be able to channel a hot aqueduct and a cold aqueduct into the palace area. Heating with steam can be a problem because the steam can quite hot, heating the rooms hotter than intended. Having a separate cold system will allow for quicker adjustment of temperature and provides a good heat sink for condensing the steam.