So I recently had an interview with an engineer and the recruiter followed up saying unfortunately we are not moving on with the process.
All the feedback you need is right there. An interview isn't a collaborative process designed for the mutual benefit of all participants (well, unless you get hired). I understand your desire to get better at interviewing, but helping you in that goal is not the responsibility of someone who has already decided to pass on hiring you.
The recruiter hasn't responded and it's been a few days so I'm not sure if he has just not seen it, or if he just is not interested in giving the engineer's contact info.
You should assume the latter. No responsible recruiter is going to provide contact information for one of their engineers to someone who has just failed an interview. The fact that the recruiter ignored your request is a strong hint that you should drop the matter. You might be perfectly polite in your request for feedback, but from their perspective it's equally likely that you're going to harass the interviewer, badger them about the reasons that they didn't like you, tell them that they didn't understand what you meant, beg for another shot, and so on. Even the most polite response could put the interviewer in an awkward spot because it's often not easy or comfortable to explain someone's shortcomings to them. Furthermore, anything they say could become a legal liability if you somehow got the impression that they didn't hire you because of your age, race, gender, etc., whether that's actually true or not.
I do know the engineer's name and after some digging, I found his business email. Would it be bad if I emailed the interviewer directly thanking him and asking for feedback?
Yes, it would be bad. Or at least, it could be. Unless he gave you his contact information and invited you to use it, you should not contact the interviewer. Doing "some digging" to find the contact info after you've already been turned down could seem like stalking behavior. I expect that that's not your intent at all, but you have to look at it from their perspective: someone who failed an interview followed up with a questionable request to the recruiter, and then didn't take the recruiter's hint and went ahead and contacted the interviewer directly.
It might seem like you don't have anything to lose by pursuing the issue, but it's best to do what you can to stay on good terms with an employer, even if they passed this time. This isn't the last position they're ever going to fill, and if by chance you did leave a pretty good impression and just got beat out by a better candidate for this job, they might call you back for a shot at something that'd fit you better. They certainly won't call you back if they feel like you were too pushy with your requests for feedback, though.
Also, ask yourself what you really expect to learn from the "feedback" that you're asking for. Do you think they'll tell you the complete, unvarnished truth? That might be something like:
After looking at your code and talking to you for five minutes, we knew you weren't the guy we were looking for: you clearly don't get what we're doing here, and you misunderstood the first three questions we asked you. You didn't look us in the eye even once, there was something green stuck in your teeth, and we found seven typos on your CV.
Of course they're not going to say that, or even ten percent of that. Instead, you'll get something more like:
We really liked you a lot, but another candidate was a perfect match for the kind of experience that we were looking for, so we went with her instead. Good luck in your future endeavors.
Here's what you should do instead:
Forget any notion of getting feedback from the company. The interview is done and you didn't get the job; get over it. This is probably not the last time this will happen to you, and the faster you learn to put that kind of thing behind you, the better.
Spend a little while giving yourself an honest self-critique. Were there parts of the interview that went well? Parts that went particularly badly? Were you dressed well? Were you prepared? Did you arrive promptly? Were you able to answer their questions? If you were in their shoes, what would your impression of you be? A week or a month from now you won't remember the experience as clearly and it'll be hard to draw lessons from the experience, so write down some notes, especially regarding the things you did well and the things you could have done better. Review those notes when you're preparing for your next interview, and use the notes from all your interviews to see if you seem to be making progress.
Practice. Find someone you trust, and who knows a thing or two about interviewing, and ask them to help you role play an interview. Go through it just as you would a real interview, and then ask them to give you an honest critique. Doing this a few times could be especially helpful if you find that you're very nervous during interviews, or if you have trouble thinking on the spot.
Here's another way to look at it: Think of the interview as a sporting event of some sort -- maybe a tennis match. If you're playing against a friend who's a better player than you are, it's fine to ask them for a critique, and they'll likely give you some good pointers. But if you lose a tournament match against a stranger, you wouldn't shake their hand at the end and ask them to tell you what you did wrong and how you could win next time. Even if you did, they'd probably say something nice, like *well, I got lucky with a few shots, good game, better luck next time." If you then called them at home that evening to ask for more specific feedback, they'd start to get a little worried about you. That's not what you intend, naturally, but if you look at it from their point of view, it's just not appropriate. So, be a good sport and a gracious loser. Evaluate your own game, and be better prepared for the next match.