So, since my husband and I have property in Michigan, we are filing a law suit with the state to get them to overturn their state wide ban. Theatrediva, have you filed suit already? If so, I applaud your efforts. That is how all these cases start, with everyday people deciding they are being treated unjustly. Who knows, one day it could actually land in the Supreme Court. Are you paying for an attorney? If you are I'd highly recommend you look into reaching out to groups like the ACLU or your preferred group *insert progressive legal action group here* who will probably pay for your counsel, perhaps put you with other couples filing suit (better chance of getting noticed and proving enough specific harm), and have experience with how to handle these kinds of cases.
If that couple resides in one of the back asswards states or territories where their union is not recognized the caregiver would NOT be eligible to take the extended leave to care for the sick spouse.
Since I'm not an attorney I fail to see how this comports with the Equal Protection Clause of the 14th amendment. You ask good questions and the answer is no one knows for certain. It will be up to Obama and his team to decide how they handle cases where someone is legally married and recognized by the federal government in one state and then moves to a state where they're not recognized. My guess is the government will give them recognition regardless, but they may not, and other administrations may act differently in the future.
Your last question is very legitimate and will ultimately be decided by courts in the future. This was not the question before the Supreme Court today though and it would have been unusual for them to have ruled on it as Edie Windsor case was limited in scope strictly to section 3 and not the other section that instructs states they don't have to recognize gay marriages other state recognized. Someone will likely not get the benefits they should have received after moving, sue, and I'd imagine, the law would be changed since it seems to be a flagrant violation of equal protection, due process, and the full faith and credit clause of the Constitution.
Again, this process is long and slow, but it's what we have to deal with. The Court isn't the ideal avenue to start a long-term civil rights movement. Ideally, this would move much faster through Congress, but I think at this point, we all know how absurd that sounds.
Scratch and claw for every day you're worth!
Make them drag you screaming from life, keep dreaming
You'll live forever here on earth.