I want to take the time to say thank you for being such a considerate person, Hrumchyk is a lucky little bird to have you looking after him.
A new cage would be better, as long as it doesn't have rust. Bigger would be good as Hrumchyk would have the space to fly and play. If you keep him in the kitchen, all it takes is one little accident and he could die. My sister came to visit me a few weeks ago, and I could smell burning coming from the kitchen, when I rushed in, I saw she was smoking a nonstick pan and I became very angry with her. Thankfully, my birds weren't hurt, but if they were in the kitchen, I imagine they would have died. So it depends on the risk you are willing to take and if you are 100% sure that no one will ever burn things or make a nonstick pan smoke.
With the cat, you can never leave the bird unsupervised unfortunately
the cat could put the claws in the cage and really hurt Hrumchyk, especially if Hrumchyk is curious. It has happened many times on this forum where a cat has killed or seriously hurt a bird even though the bird was locked in the cage.
All the best, Oliagub! I look forward to the pictures of Hrumchyk happy and healthy in the future.