I realize this may vary depending on the production value of the game and if the game is free or not. I’m starting a project as a level designer and am tempted to buy models but I thought that would make the game look cheap. This isn’t a matter of model quality though. I’m aware that there are good and bad models out there. What I am worried about is that the models would stand out and people would recognize them as other people’s work. I looked around but could not find much debate over the subject.

No, it's not frowned on. It's fine for the models to be recognized as the some ones in other games if both games bought them legally.

No its not frowned upon. Dozens of hames on XBLIG use graphics from 3DRT. Lots of games on iOS use graphics from the Gamesalad and Unity asset store. There have also been a few relatively high profile indie games using the oryx sprite pack.
There are also several games using textures from cgtextures.

I think you are over-estimating the general public. Most people just play games, they are not at all involved in the game dev community. I would be shocked if you even recieved a comment about using stock material outside of here or another form of dev social media.

There's no shame in using stock models and stuff, plus, when you start building a library of your own models and stock stuff, you can start doing kit bashing. Basically this means you take parts from the various models and put them together to make new ones. Like if one of your models has very well modeled hands, which is a pain to model up and get good edge loops, you can just use that hand for all your other models, making minor adjustmenst if need. Same goes for props, clothing, etc.