Muslim is not allowed to deliver unlawful (Haraam) items at all. If you are sure the job will definitely make you carry or deliver such Haraam items even in any small percentage, you will not be allowed then to do such job.
Non Muslim seller who claims that the meat he is selling or wants to sell is Halal can not be taken as an authentic evidence that the meat is really Halal. If Muslim tells you that his meat is Halal and you have no reason to doubt his statement, then you can take the neat as Halal.
Restaurants selling alcohol or pork can never be trusted as utensils and plates used in cooking or serving alcohol or pork become Najis and and every other food put in Najis utensils and plates becomes Najis as well.
Usually we deal with questions related to practical issues which people experience in their day to day life. Your question looks like very far from day to day life, nevertheless if you want the answer for knowledge, it is obligatory to save our life. Deliberate failure in saving life and causing death is a sinful act. It is not permissible to cause death to yourself even if the only food available to remain alive is pork, but no more than what make you survive.
Allah knows who is entitled to be forgiven according to the degree of his sincerity and real decision not to commit the sin again. Allah (SWT) said in Quran (Say, O My servants who have transgressed against themselves (in committing sins) : Despair not of the Mercy of Allah, verily Allah forgives all sins, truly He is The Oft-Forgiving, The Most Merciful) Sura 39, verse 53.
We need to always seek forgiveness and repeat that whenever we remember our sins.
Muslim is not allowed to sell or take part in marketing or selling alcohol or pork. You should refrain putting such unlawful items on your website or platform. If you sell such Haraam items,bayou will be committing a sinful act, and every income from such work is Haraam as well.
You should keep your platform and business activities clean and pure from any Haraam business.
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.
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.
Muslim is not allowed to feed any one pork oe any part of it, even to those who usually eat it.
Feeding non Muslims Non Halal meat is a matter of debate between the scholars. I advise you to refer to your Marja' of Taqleed.
As jurists say, it is prohibited to buy or sell pig meat, but as for working in a place that serves pork, it is permissible. The problem would arise if you yourself were to prepare or serve the pork, in which case most jurists say it is of obligatory precaution to avoid.
You would need to refer to you Marja' Taqleed for the specific fatwa on this issue.
And Allah knows best.
If it is your house, then you must have authority over what enters it and what happens inside the house. Instead of getting to the stage of disowning, or expelling them from your house, try to get them to compromise.
For example, ask them to cook their non-halal means outside.
Also, continue to do as much dua for them as possible, that Allah ta'ala enlightens their heart and guides them. Remember that the dua of a parent is always answered.