Were there any vertebrates with 6 or more limbs?
I've not read it but this SE question Why don't mammals have more than 4 limbs?
though not a duplicate (it asks about mammals not all vertebrates) may have some elements in its answers pertinent to your question. 

And the question Why are there no vertebrates with more than four limbs? has been asked in Quora where it has 7 answers you may find useful.

Also this discussion thread Why can insects have 6+ limbs, but mammals only have 4? in reddit.


  Does paleontology has any evidence of a vertebrate that had 3 (or more) sets of bilaterally symmetric limbs or lobe-based fins?


You sent me far down the rabbit hole with this one, for reasons that escape me my instant response was 'yes the Coelacanth' but as you correctly point out that has only 4 paired fins.

As seen from The origin of tetrapods our limbs came from fins, Coelacanth is a good example of the likely early beginnings of Tetrapod limbs, they occasionally use them to 'walk' on the seabed.

The image provided in this National Geographic article Coelacanth Rediscovered gives a much clearer picture of it's 'limb' structures.



Latimeria chalumnae, the coelacanth (SEE-lee-uh-kanth), thought extinct until a biologist identified one off the coast of South Africa in 1939.

I've trawled Google for early chordates looking for examples of more than two pairs of fins &amp; haven't been able to find any, it seems the earliest paired fins appeared in (or shortly before) Ostracoderms &amp; from there continued in an unbroken line to the tetrapods &amp; our limbs.

Paired fins don't appear to have developed in chordates before then or to have ever developed into more than two pairs.

So to the best of my knowledge it appears that the answer to your question is no.

Other links.
Yunnanozoon
Myllokunmingia
Haikouichthys
Evolution of fish