This is not true. What is true is that, as Frauwallner has shown, there is a core of material that is shared by the Theravada Abhidhamma and the northern Abhidharmas. See more here: Abhidharma - Wikipedia
Not really. There is continuity of course, but the Abhidhamma system is a later development. The best evidence for this is the very fact that while the early suttas of the different schools are often more or less identical, the Abhidharma texts differ considerably. This shows they developed after the suttas and thus are not from the time of the Buddha. For more info I would say read Frauwallner’s study of Abhidharma or this essay by Noa Ronkin: From the Buddha's Teaching to the Abhidhamma | Cairn.info
The evidence is in the works of modern scholars who have studied Abhidharma like Frauwallner. Of course as I have pointed out above, the best argument for the lateness of the Abhidhamma is the differences among the abhidharmas of the different schools.
I do not think they are necessary to understand the Dhamma, no. That does not mean I think they are totally useless or wrong however. Though I do think they contain ideas that were not taught by the Buddha, for example, the theory of momentariness or the theory of the “heart base”. So it can certainly confuse people.
I think I mainly agree with Sujato’s position here. The commentaries are very useful, especially for translators and scholars, but they are not the word of the Buddha and must be read critically. That doesn’t mean however they they can’t help illuminate and understand certain passages, so I wouldn’t say people shouldn’t study them.