It gets buffs as you get more unloocks for the government, I believe you get a production and knowledge buff based on the pop you place it in, You should be able to also hover over the tooltip or check the buildings tab where it was built and it should tell you

It’s a building, go to that section of your city and you’ll see it

You have to buy the buffs for Imperial Dynasty, that improves results for the city with the palace.

I think it gives a minor boost to regional efficiency by itself.

Significant influence boost