Regarding your question toward the end,
It's more than things in common that indicate common ancestry. It's the synchronization of things in common as well as differences across several fields of independent research. I've said this before but, the fact that people can take DNA and use it to predict the spacial and vertical location of a fossil in a remote part of the planet, tells us that our genetics and the fossil record are 1 to 1 in how they match up.
That is to say that dna changes subsequently change our morphology which results in the creation of the fossil record. The fossil record shows us our lineage and that's that. Simple as. Biologists suggest that the fossils are complimentary to what they've already established. As a geologist, I would say that the theory of evolution simply reinforces what we already know about fossils. Both independent , both both telling the same story.
And differences, are mutations. Mutations are those differences. Cladistics is based on morphological differences and genetic differences. So you can look at how similar things are and how different they are, but you come out with the same cladistics based phylogeny which repeats itself in each independent field of study.
So the differences and similarities both support common ancestry.
As far as what would not support common ancestry, things that do not fit the pattern would contradict common ancestry and would suggest independent creation. For example, a dolphin found in the ordovician would contradict common ancestry. A reptile with mammary glands would contradict common ancestry. A bird found in the devonian would contradict common an entry etc. A fish being genetically more similar to a human than to an amphibian would contradict common ancestry etc.
These are the things that, if you presented, It would probably break the theory. But after 200-300 years of uncovering fossils and sequencing genomes, we come to find that these things, to our knowledge just don't exist. I certainly haven't found a dolphin in any paleozoic rock that I've looked at. Or any equivelant. And I've looked at probably more rocks than...well I'm geologist, I've seen a lot of rocks.