Regardless of what she was doing while she was there, the only relevant thing that is going to be considered in an unemployment hearing is the reason that was given for her termination at the time she was terminated. If the company had a problem with her doing other work while she was on the job, they should've warned her to stop doing it, and then if she did not, they could've terminated her for that cause. That would've been a very deniable claim. But in truth, it sounds strongly like they had overhired, they needed to let someone go due to their staffing needs, and they did, according to what they told her originally, let her go for that reason.
Then, later, when they realized that her receiving unemployment insurance was going to cost money, they used their option to appeal the decision to approve the claim to try to bring up all this other stuff, like how she was always working on some other type of project while sitting at her desk on their time. When, really, if that had been the issue, they should've said something right then.
And the OP should not neglect to go to the hearing, where, in my opinion, she had an excellent chance of being upheld and continuing to draw benefits. I have actually seen cases where the ex-employee was later discovered to have committed theft or embezzlement while on the job, yet because they were laid off cleanly for lack of work, not terminated for misconduct, they were entitled to keep on drawing. As we had to tell these very disgruntled employers, you can press charges against this person, but until they are actually incarcerated, and no longer able and available, you cannot stop their claim after the fact.
If the OP does not go to the hearing there is a much better chance the employer will prevail and the decision to grant benefits will be denied. If they are declared ineligible, they will find themselves overpaid for the amount of unemployment they have already drawn out. And that is never a good thing, though it is not a fraud overpayment, it is definitely still on the record books and they will eventually want that money back.
I wonder how this one came out. As I said, I think the claimant had a fairly good chance if they showed up and adequately tried to defend their eligibility.