Can a university claim patent rights for inventions performed on my own time?
Note that a detailed response to your question, as well as any answer tailored to your specific case, will need to be given to you by a lawyer. And, if you think of doing anything that could even remotely be considered borderline or controversial… just ask a lawyer. And even if you don't do anything borderline, better safe than sorry!
Now, I want to explain some of the background of that answer: I have acquired some experience of my own, though not in the US, on dealing with being employed as a full-time researcher and professor, yet maintaining a certain level of activity outside of my employer. I actually produce intellectual property on my leisure time (both copyrightable materials and patent rights), and have consulted with law professionals on how to effectively keep control of my own IP (as opposed to that which I create for my employer).
The main conclusion is: you actually have to be very careful to be able to fend off any potential claim of your employer on the IP created. Of course, it's a risk of the “if it comes to that” variety. If your university is not interested in pursuing any claim to such IP, you may never have a problem, even if you did not clearly separate your own IP from that created for your employer. But… you never know what may happen in the future (e.g., your current understanding with the dean may not hold in a few years), so you should take maximal precautions.
In short, you should assume the following:

Everything you do on company time is owned by the company. I'm pretty sure that this includes your “20% time”: if it's 20% of the time the company pays you, it may mean that they leave you free to do research in a direction you choose for yourself, but it is still research you performed as their employee.
You certainly can't expect to be paid 40 hours a week, work 32 hours for your employer and then spend the remaining 8 hours for someone else, can you?! (in this case, the “someone else” is yourself as a freelancer)

Don't use resources from your university. This is also very typical of contracts and, at least the few European countries I know, the legal status quo: use of company resources means the company has rights on your production. This includes every resource you list, even smaller things you may not think of, like email account, FTP/web/ssh hosting, internet connection, …

Everything you create while inside the company is not yours. Walls, internet connection, desktop, electricity… are company resources. While courts in some jurisdictions have recognized that certain limited activities done inside the company, such as reading email during lunch break, may fall within the personal sphere (and not the professional sphere), it is not likely to be the case for any larger activity.

Finally, if you think of doing on your spare time anything that is thematically closed to your day job, take care: it becomes hard to claim that your own activities are not linked to your university position and resources, and do not in any way interfere with it, if you are a computer science researcher by day and patent new sorting algorithms by night :)


And, did I mention? Get help from a professional, aka lawyer.