First of all, congratulations on graduation and successful completion of the NCLEX. Sadly, it has been....well..years since I applied at both Rush Memorial as well as what was formally UIC; therefore I am unable to supply you with any information that is both relevant to their current practices as well as the overall current socio-economic conditions.
"2a. When I am sending U of I Health my resume, is it considered bad manners to send a couple of letters of recommendation along with it?"
First of all, the fact you labeled your question "2a" is just ridiculously adorable. Perhaps it is the lack of sleep or decent coffee, but I giggled like a fool and pointed at my computer screen. I suppose this is as good a time as any to note that there is no one else around. hmmmm. ::cough::
The thing to remember is HR is inundated with resumes, CV's, inquiries, etc. Therefore, as much as you want to tout your excellence, sending anything additional without having it requested is more than likely not going to earn the attention you desire. As always, offer to enclose letters of recommendation upon request. I have found it is much more effective to have a strong cover letter, well phrased resume and a politely worded phone call to the HR office offering them any additional information when they review my resume.
"3. Is what they told me in nursing school accurate, about a resume being an absolute maximum of two pages long? (As I have 20 years of work experience, most of which correlates to the nursing profession, I am concerned about chopping too much off)"
Yes, it is very true. In fact, ideally the resume should truly be no longer than a single page though sometimes the length simply cannot be helped. Now, I understand your concerns regarding your work history, however, each job truly only needs three or so concise bullet points with active tense wording (as an aside, jobs at which you stayed for considerable length tend to have more bullet points--that is both expected and acceptable). Be sure to highlight traits of your previous jobs which can apply to nursing. As a sub-point, keep in mind that potential employers do not want your complete work history. ::arches eyebrow:: After all, what could my first job waiting tables at the tender age of sixteen possibly tell them?
Re: Signing your name with an RN: That "new car smell" will wear off after a while. Embrace the newness and all the glorious dorkiness that goes along with it.
I'm not sure I was much help, regardless, here are my ramblings.