The domestic cat is a small, typically furry, carnivorous mammal. They are often called house cats when kept as indoor pets or simply cats when there is no need to distinguish them from other felids and felines. They are often valued by humans for companionship and for their ability to hunt vermin.
If neutering the male cat avoids you or others expected problems, then it will be allowed to avoid that expected harm,
It is not allowed to touch the name of Allah as well as Quranic verse or part of it with out Wudhu. Names of the Infallibles are also included according to precaution.
Cat is not Najis but we must keep the due respect of the name of Allah in every possible way and in every situation.
Yes if neutering the animal will avoid you, your family and others a possible harm.
If she is spraying (urinating) on you/your clothing, and you have cat urine on yourself or your clothes, you should wash out the urine before praying. If not, it's not a big deal, but it might be annoying.
Yes it is allowed to have a cat and touch her. Praying Namaz needs your body and cloth clean from the hair of can or any animal which is not permissible to eat. Ayatullah Sistani allows few hairs.
Cat is a Taahir animal and we have many Hadeeths from the Prophet (SAWA) and his Ahlul Bayt (AS) that keeping a cat at home is allowed and performing Wudhu from remaining of water from which cat drank is allowed. About the name of a cat in the house of the Prophet (SAWA), the narration is found in Sunni books, but I could not find it till now in our Shia books. It make no difference as the Hadeeths which we have are authentic in allowing having a cat at home.
It is fine to have a domestic cat, and the Prophet (S) is said to have been favourable to cats.
Some cultures have fear of cats due to superstition.
It is allowed to feed the animal non Halal food or Najis meat as the animals are not responsible to avoid non Halal.
You need to be careful that your home or clothes do not get Najis from that Najis food.
It is not allowed for you to kill your cat just because of her illness or sufferings. Allah, The Most Merciful, Who created all, is more merciful on her than you and He terminates her life whenever it ends according to His Mercy and Wisdom.
You do not need to get rid of the cat. The bowl and all items are not Najis unless you have an evidence that it had wet Najis item in it. If you know that it is Najis, then just washing it with water will make it Taahir.
Thank you for your question. Your regret is sufficient for repentance from any wrong doing in this situation.
May you always be successful
I am sorry to hear about your loss.
It is narrated that the Prophet (S) said: “Keep many domestic animals (al-dawajin) in your houses so that the shayatin (demons) are occupied with them instead of with your children.” (See Tibb al-A'immah)
I think there is some truth to this.
Of course, maybe your cat just died and there was no other reason for it. The lifespan of animals is also in the hands of Allah, and this life is transitory for all of us.
At least you got to enjoy some time together, and one thing I have always admired about animals is that they handle death with grace and acceptance.
With salaams and du'as