Wanting help to identify style and how to properly care for wood veneer  

I had one of those tables---got it for $5 at an auction. The veneer on the top was badly bubbled. I thought it would be useful but you kind of need to get down on your hands and knees with a flashlight to see what's in it! But it's rather handsome. The wood is walnut and the door inserts are pre-woven cane sheets, not parchment. Not sure what the finish is---probably lacquer. It is by an American manufacturer---I can never remember the name of the one that did a lot of this stuff. Google "walnut cabinet cane doors" and you'll see plenty of examples, including Knoll and McCobb but that's not what yours is. I also had some tall cabinets by the same company that were pretty nice.
The bench is a very common item. Count yourself lucky if yours is not wobbly. They seem to be pretty prone to loose joints. I am not crazy about the stained wood look but these are popular with people just getting into MCM. I'm not sure about the finish on this one, either---sorry!

That's actually very helpful. Thanks for all of the info! Surprisingly, the bench is solid, no creaking or wobbling. The slats and supports on the bottom are fairly chunky with good quality hardware it seems. Thanks again!

Give them a good rubdown with either mineral spirits or naphta and clean rags followed by paste wax if the original finish is in good shape. Both are good for removing old wax build-up and grime, and neither will harm existing finishes or raise the grain of bare wood.

The bedside table things or whatever they are remind me of Founders Furniture.

I was just coming here to say the same thing, having finally remembered the name. But I googled it and couldn't find these end tables in particular, and I did find pics of the Founders name in huge letters inside some drawers...so seems like they were pretty good at marking their stuff?
I think there were a bunch of places that made this stuff, though. My tall cabinets had very small black metal knobs and not inset wood pulls like on some things I've seen in this style, fwiw.
WOW---i found a photo of them. That is the least amount of time I have ever spent hunting for a photo before finding it. I should quit while I'm ahead.

Thanks, guys. Really helpful info, I do appreciate it. I went over everything with a damp cloth and light soap/water. Everything cleaned up pretty well. Not really sure if I want to do anything more, but I'm really enjoying the pieces.

I have owned some Founders Furniture pieces that were not marked, so anecdotally the company was not 100% reliable.