there is no conviction or anything else "supporting" that accusation
The First Case
How did the other court came to the conclusion "the jury had found he
digitally penetrated her"? Are there any physical or technical
evidence supporting that?
There was no conviction. But, there was other evidence supporting that accusation. This is why she won the case. The Wikipedia article on the case summarizes the evidence in detail. She had, for example, a dress with semen on it, that she was wearing that day. Also, sworn testimony is evidence that provides a basis for a jury verdict. Physical evidence is not required.
So Trump is convicted of a sex crime in a civil case!? How can a sex
crime be handled as a civil case? Aren't the burden of proof
completely different in civil cases and criminal cases? In civil cases
you just need your claim to be more likely than the other party's
claim ("51 %" so to speak) while in a criminal case it must be beyond
reasonable doubt.
Many things that are crimes, including rape, can also be the basis of lawsuits for money damages. If you are convicted of a crime beyond a reasonable doubt you can go to prison. If you are found liable in a civil case but are not convicted of a crime, you just owe money damages and don't go to prison. One of the most famous examples of this is the O.J. Simpson murder case where he was acquitted in a criminal trial of murdering his wife, but was found to be civilly liable of murdering his wife.
The jury in the first lawsuit found that her accusations were true, i.e. that Trump raped Carroll. Thus, his claim that she was a liar was false. Falsely accusing someone of lying is defamatory. Accusing someone of rape is only defamatory if that isn't what happened. But, in this case, the jury found that he did rape her.
Also, Trump did countersue in the first case, accusing her of defaming him, and lost. He lost because the jury found that her accusation was true.
Statute Of Limitations In The First Case
The statute of limitations issue has two parts.
First, New York created an exception to its usual statute of limitations for civil lawsuits arising out of old rape cases in a narrow window of time that she used.
The criminal statute of limitations of rape cases had expired, so no criminal charges could have been brought at that point.
Second, the statute of limitations for defamation runs from the time that the defamatory statement is made. Trump raped Ms. Carroll many years earlier. But he accused her of lying shortly before her lawsuit was filed. So the statute of limitations for the defamation lawsuit against him had not expired.
New York does not have a criminal defamation statute
The Second Case
In the second case, after the jury in the first case found that Trump raped Carroll and then lied about Carroll being a liar, Trump was sued because he lied again about Carroll being a liar after already losing the first case. Carroll promptly sued him again for the new defamatory statement.
Even while the second case was pending and during the second trial itself, Trump continued to lie and say he was falsely accused.
In the second case, the first case's jury verdict was binding on Trump. So, the only issues that he to be shown in the second case were that Trump repeated his lie about Carroll falsely accusing him, and her damages.
The open and shut nature of the case when he defiantly lied about what happened after a jury had already found otherwise in the first case, is part of the reason that the punitive damages award against him ($65 million) was so high.