No you don't. I've not had to pay anything for the Pro adjustments related to Clarity, Detail, Impression, or Simplify.
We don't know that since Topaz hasn't provided a roadmap for updating all the plugins into Studio. I'm assuming they'll continue to add all the remaining plugins with the probable exception of PhotoFX which is kind of what Studio replaces. With all the benefits they get on the development and maintenance side of things, it wouldn't make sense to only update some of the plugins unless some plugin was just going to be effectively discontinued. Again, that's my assumption/opinion. And if that's correct, they will all upgrade for free. Of course, you'll have to wait for the upgrades to be implemented; in the meantime you can access the plugins via the plugin menu.
I'm a LR user. I edit in Studio from LR. I call all the plugins from there and use the free adjustments and those Pro adjustment that have been upgraded for free. I used to use PhotoFX in the same way.
I don't have PS. So one benefit is the layer support in Studio. It's also much easier to experiment with many plugins; I can create several different images and version all in a single session without creating an excess of TIFFs back in LR. If warranted, I can save my work in Studio as a standalone project outside of LR and return to continue the work. I can also return multiple images back to LR, e.g. I can return separate black and white and color versions and continue to process them independently back in LR (this requires a little trick that takes advantage of the way Studio saves back to the host program.)
When the plugins get migrated into Studio, the individual panels in the plugins become a single adjustment. So it's easy to use the specific features of different plugins and have each treated as a layer which can be blended and masked as desired. This will also eliminate the redundancy that exists among some of the features of plugins.
There are certainly other advantages. And these are in addition to the implicit benefits of having improved development and maintenance costs. Especially since Topaz is a very small company.