HI, Thanks for your reply. I am going to address your concerns separately but you can as anything after that.
Is it possible to keep the house clean enough from cat dander also that we can peacefully coexist?
I am sorry but it's very very difficult. almost impossible because dander can spead in the air and can go to any part of home. Moreover the first treatment of allergy and i would say the last also is to stay away from allergen So the cat has to be away from you. I am so sorry because i do understand your family concerns but for your safety and heath one has to go out and that will be the cat i guess :) There can be one more scenario that your current allergy might not be cat related, so if you can shift the cat to some friend's place (if possible) and see if this helps your symptoms. If it does then this will confirm that the cat is the cause and you can further act accordingly.
If not, once the cat is removed, how long will it take for my allergies to calm down?
It can take two weeks or more because the cat dander will take sometime to get cleaned. However you will start feeling better in three to four days prvided the cause is cat. Yes your doctor can check you fr cat allergy but you need to see an allergist for that.
I would like to share with you a home remedy for allergy which works in most of the people. Mix one tablespoon of black pepper and one tablespoon of ginger extract in a mall bottle of honey. Make solution and take one table spoon twice daily.
I wish you good luck that you get well soon, please ask if you have any question because our discussion always remains open so you can always ask more and I will be glad to assist you. Your excellent feedback is requested and will be appreciated, regards.