I have a Kia Rio (only a few years old). I was driving along on a winter day, not seeing the camouflaged pothole that lurked under the snow, until I hit it. I was only traveling about 25 mph but still, the car banged and tire went instantly flat.
The tire was fixed and the mechanic said that the rim hadn’t been bent.
This drama happened over a month ago and to me there seem to be subtle changes in the way the car handles and idles; to me, it seems idle rougher and ride rougher than before. However, my husband hasn’t noticed anything different. It’s possible that I’m hypersensitive to all the rattles and bumps now, or that he just doesn’t notice.
My question is - running into a polehole hard enough to make the tire go instantly flat but not hard enough to damage the rim - what other possible damage could have occured?
I think you have 2 unrelated circumstances. Yes you may be hypersensitive but that does not mean it is wrong. I would have the alignment checked for handling problems, then look into what routine maintenance you might need such as air filter gas filter etc. It should all be in the manual.
It is very possible that the suspension has been damaged after hitting a pothole. I see it often. Have a shop, an independent shop or a dealer, put the car up in the air and search for bent control arms, spindles, etc. and also get an alignment.
But, it is possible that the rough idle was already there. When something like this happens, vehicle owners often tend to pay attention to everything and then notice sounds that were there all along or at least for awhile.
It’s possible to suffer suspension damage that will not be noticeable to the naked eye. Something bent, caused the strut to fail, etc. The usual suspect is a tweaked lower control arm.
The car needs to go on the front end alignment rack.
In theory, a pothole should not cause a performance problem. In practice, something could happen. Vacuum hose dislodged, vacuum nipple broken, etc.
It could also be that nothing has changed and it’s just hypersensitivity.
Many years ago I was in a major car accident and my car was a total wreck; and a mangled wreck at that.
The engine used no oil previously and was not touched during the wreck. I sold this engine to a gentleman who installed it and that motor burnt a full quart of oil every 5 miles. It smoked like a grassfire; something it did not do before the wreck.
Apparently the impact jolted the crankshaft forward which in turn cracked some piston rings.
So my opinion is that anything is possible. 
Living in New Engkand I can testify that hypersensitivity after hitting a pothole is normal. I can also testify that hitting a pothole hard enough to cause a pinch flat can do damage and/or cause the alignment to go out. And yes, it can knock something loose that can affect the engine operation.
Have the Kia checked out. It’s the safe thing to do and you’ll sleep better.