Do they sell men turtleneck sweaters below $20?

For women's, so might be same for men: If it's under $20, it's thin to comfortably wear under a sweater. Keep in mind for tbe white one, the neck part will look normally thick, but the body part will be a tad transparent, which means wearing it under something always.

Check Nordstrom Rack

Yes

You could call ahead to check but i would guess nordstrom racknis a better bet than Michigan Avenue.

I'm not sure about the Michigan Avenue location but Nordstrom Rack does. Maybe try the south loop location. Of course they need to still be in stock, not really a summer item