I'd have concerns as well.
First off, if you ever went to sell the house down the road and stuff isn't up to code, the buyer can insist that you bring it up to code before sale completion. So then basically you're doing the work twice. I have had enough contractors do work in my house that I would have serious issues with this kind of shoddy workmanship. I don't think it's micromanaging – contractors, even good ones, will try to cut corners wherever they can if they can get away with it. I have had contractors price us a job, come in and start the work and then realize the job is much bigger than they accounted for – and start cutting corners so they can still make a profit.
Finally, if they don't respond to your questions, I would also consider this a red flag – get in touch with whoever your foreman is directly, by phone and talk it over. Often, the lead guy subcontracts out to someone new every week and doesn't really have an eye on every single contract they're working on.