Xolair is an extremely expensive medication and considered a last resort for patients with a medical necessity (per a letter from your doctor to the insurance company), or sometimes the drug manufacturer may offer reduced/free programs to obtain it. Unless your doctor has a very good reason for picking Xolair ( which also can take 6-9 months to reach full efficacy), other options could help more quickly and be a lot more cost-effective.
Xolair is still a very new medication in the biologics family. It's generally quite safe, but as it is so new there are updates your doctor/oncologist receives to let them know the latest studies and their results regarding Xolair and CLL. Xolair is most generally used to treat severe, corticosteroid or other typical asthma-medication non-effectiveness. It does so because it decreases a patients IGe levels - it also leaves ones immune system more susceptible to infection. It's a great medication, but not known to be a typical CLL choice. It does help with allergic reactions too, but again, it's an extreme choice if all other options haven't been tried, or ruled out due to your personal health history.
Though the below biologics mentioned may not help your dog allergy, they're worth investing regarding your CLL:
Another biologic studied with a focus specifically on CLL patients using the biologic, Lumiliximab was found to have low instance of serious side effects and offered improvement for most patents - you may want to ask your doctor about this one because of your CLL.
There's also the biologic, Alemtuzumab, which is another specifically for CLL patients.