Minwax's polyurethane or polycrylic will be just fine. Polyurethane is an oil-based product. Polycrylic is a water-based product. Sometimes polyurethane will "yellow" over time and polycrylic is less likely to "yellow".
You need to knock the shine off at the very least so there is something there to adhere too. I would use the same treatment as you did to the doors to insure it all matches. Real wood is always better than anything else.
You will want to us a polyurethane over the stain for sure. If you don't it will leave the wood susceptible to moisture and also won't allow for cleaning up if something is spilled on it. Liquids will just soak in and more solid spills will be very hard to remove. You will want to use at...
