State tax filing

Married Filing Joint is probably your best filing status, both for Federal and California.  I'll get back to more on your Federal return shortly, because you may have additional deductions you're not aware of.  But suffice to say that you have greater access to deductions and credits, you avoid the community property headache, and, if your income and deductions are high (which they might be), a more generous treatment for the dreaded AMT (Alternative Minimum Tax).

Having said this, let me answer your direct questions:

1. Yes, you are correct.  It could be splitting hairs a bit because many states will call you a resident if you had a physical presence in the state of 183 days or more and maintained a residence.  She maintained a residence in Illinois, but may be close on the physical presence (especially if she was back and forth a bit).  But you will run into major complications if you must file an Illinois resident return since she is also a California resident (where her main home, or domicile, is).  You will want to prepare the Illinois return first, as this FAQ explains:  https://ttlc.intuit.com/replies/3302052
2. You will want to file Married Filing Joint in both California and Illinois.  California is your state of residence for the two of you, so they are going to tax all of your income no matter what.  You won't gain an advantage filing MFS for that reason, and you have listed a few others why it might not be a good idea.  Illinois, likewise, requires that you file a joint return in this circumstance.  They would allow for a separate return if she is filing a resident return which is another reason why you want to file the Nonresident return (it's a lot more work to do the resident return).  How Illinois will calculate the tax is that they will look at what they would tax your entire income as Married Filing Joint and then calculate the amount of tax proportionate to your wife's Illinois income.  But, as a corollary to this, yes, California will give you a credit for the amount of tax you pay to Illinois for the amount of income earned in Illinois.

Now, for the Federal return, there is another potentially large deduction you may be entitled to.  Since she was working in a temporary location, the IRS does allow for some substantial deductions for those who are in these situations.  Any deduction claimed must be reduces by any reimbursements paid by her company.  For a full consideration of this, please see this IRS Publication:  https://www.irs.gov/publications/p463/.  Chapter 1, Travel, and Chapters 4 and 5 you'll find especially useful.  Take the time to go through it so you can claim everything you are entitled to.  Feel free to ask more questions and I'll be happy to assist you with them.  To enter these expenses in TurboTax, you will use Form 2106, described in this FAQ:  https://ttlc.intuit.com/replies/4800418