What name of brand walnut oil can I use to recondition my walnut wood cutting board? There are some many on the market.

Walnut, or any other wood, does not need oil. We add oil only to improve its appearance. So the frequency with which you apply it is entirely up to you. There is no need to use walnut oil on walnut wood. Look for an oil that will not go rancid quickly. Canola is fine.
If your board is highly scratched or stained, it can be sanded. Find a willing woodworker or handyman. For DIY buy a sanding block and an assortment of sandpaper varying from 80 grit to 180. Start with the low grit and move up. Apply oil after sanding is done.

How often to recondition walnut board with oil (once in a year, every month or every week)?

How long does it take to recondition a walnut board? Any particularly recommendations to do it faster.