i'm not sure that a change of counter will neutralize the tone of the oak, but for sure getting rid of that rust paint will help! I'd use the sample on the right and a warm gray wall paint.

Paint the walls a nice neutral color and use the counter material on the right , but really painting the cabinets would be a huge improvement.

You may need primer to cover the red wall paint, so start there because the red is reflecting onto the cabinets, increasing the orange tone. Prime it to white or cream and then pick a neutral color for the walls when you are done with countertop and backsplash, if you do a backsplash. Lighter countertops would help because they will reflect light. Dark countertops will work, but they often darken a kitchen, so be sure there is good lighting before going dark. Also, change out old fluorescent light bulbs (curly Q bulbs) if you have any and play around with different levels of LED lights. They come cool to warm and will affect the tone of the cabinets.

It is not necessary to paint the cabinets, though it looks like from the side of the cabinet that painting has begun?

This is my oak kitchen with light countertops and wall paint:

https://www.houzz.com/photos/my-pics-work-in-progress-phvw-vp~93784356

Here is another refreshed wood kitchen:

If you go with a dark countertop, choose a light, simple backsplash:

Eye bleed time. NO! Paint the cabinets. They bleed into the floor and bleed into the walls.

If the cabinets stay, the last thing you need is a dark counter with more orange in it. You need light and creamy. No grays.

If the cabinets stay, I agree with light and creamy.

The one on the right. It blends well with the cabinet color, whereas the one on the left doesn't blend and would accentuate the redness of the cabinet.

Paint out the cabinets, and the walls. Leave the rest until ready for major change.

I would start with painting the walls a light neutral color & a lighter countertop. I would paint the cabinets only if I could afford to have it done professionally.