Some airlines do that. Majors are normally pretty thorough. There are also some common-sense checks they can do such as TT= ME+SE (assuming no helo, glider, etc).

I would not go to an interview with any obviously detectable errors. If you miscalculated some night or IMC, they might never notice.

No, same thing happened during to another guy during my interview. He put on his application that he had never failed a ride but the first thing they did during the interview was look through our log books. A look at his showed he had failed one and he was sent home right away. The lesson is- disclose everything because they will find out sooner or later and if they find out later well... you'll go to piloting something with 18 wheels.

Be very careful about making such assumptions. While I may agree with you in very broad terms, you just never know what these folks know or what their background might be. It’s more important to recognize that they hold more cards than you do. The HR folks who interview you do so for a reason. It’s imperative that you give them respect… So be careful about lumping groups of people into one category.
Requesting training records from other airlines is part of the PRIA. They don't come from the FAA, correct, but they are still covered under the PRIA. Airlines are required, by law, to request such records and subsequently furnish such records. So airlines don't have a choice when it comes to requesting previous training records.
You can request your airman file from OKC here: Get Copies of Airman Certification Records
It sounds like Eagle is paying very close attention to logbooks. From their perspective: if he lied about this even before he’s hired, what else will he lie about? There really is no margin here – if someone lies on a legal document (the application) and it is discovered, they’re out. In this case, he’s lucky he wasn’t hired and then it was discovered. He would have to explain why he was fired for the rest of his career.
This actually brings up a good point. If flight times don’t add up it may just be an addition error in the log. But if on the application the stated hours are more than what is reflected in the logbook we have a different issue. Never overstate your hours. Yes, it can get you booted from an interview. Overstating hours is a sticking point for anyone, but some airlines are much more harsh than others.
Typically, when a logbook is reviewed it is a spot check. They’ll open random pages and yes, calculate the numbers. If there seems to be something wrong they may take it farther by continuing page by page until they understand the discrepancy. If something doesn’t make sense to the “auditor” they will investigate until they are satisfied. Important note: It is in your best interest to write legibly and cleanly so that anyone can pick up your logbook and know your history. If an interviewer can’t logically “follow” your logbook how would they end up with a good feeling about you?

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Wow, I can't believe there are six pages of debate on this subject. Folks, I think you're missing the bigger picture. Airlines ask about previous failures because they're interested in whether or not you take responsibility for your actions. The fact of the matter is that you are a better pilot because of the mistakes you've made. It's not a bad thing. Take ownership and talk about what you learned.