Honestly, not a lot would happen in the general scheme of things. But, to answer more specifically, we have to consider a couple factors.
How did they get there? Was this the result of humans reviving them a la Jurassic Park? If so, they'd have to have been altered significantly just by the act of de-extinction. They retroactively said the same of JP's "dinosaurs" in Jurassic World. This scenario is a whole kettle of fish on its own ...
If it's just some mysterious time slip or other unintelligent phenomenon bringing literal paleozoic creatures from their time and environment and dropping them in our modern oceans, it's going to end tremendously poorly for them.
For one, they'd have no natural immunity to the modern bacterial and viral ecosystems that have mutated severely since their time (much more so than multi-cellular life has in the same amount of time). So, much of it would die from infections their bodies would have no clue how to combat.
(Before editing, I stated that climate differences would be a huge problem, but I've had it pointed out that I missed you stating they were adapted to our climate so I take this back)
Finally, ecosystems tend to compartmentalize by niches. Creatures in a given environment succeed because chance and passing on of traits has specialized them to fill one or a series of niches in the environment. While these creatures certainly filled such niches in their time, they are filled now by other creatures, most of which are far more suited to the changes in the environment I discussed above.
So the ultimate answer is going to be this: I still stand by my conviction that most of them would die off very quickly, even without climate being a problem. Though I reduce this to 78% or so, versus the 99 I originally stated. Though one comment was right that larger animals may find it easier to bludgeon their way into already-occupied niches, though I worry about competition there still giving an edge to established modern life.
I want to add, the giant dragonfly really -can't- be adapted to our modern environment, because insects simply cannot be that big with the oxygen content of our atmosphere. That's the primary governing factor in why things like that are gone.