My wife of 4 years has just announced she wants to divorce me and return to her native country 10,000 miles away with our 1 year old and 3 year old son.
Prior to our marriage, she engaged in a sham marriage in an attempt to obtain a green card, but was rejected because the individual she paid to marry her was unemployed, and did not meet sponsorship requirements.
Unaware of this, I asked her to marry me when she became pregnant with our 1st child. That's when she confessed to me her existing marriage, which was in the process of having a divorce finalized.
I also feel she has seduced me and induced me to enter into yet another sham marriage.
She is a loving but incompetent and potentially dangerous mother. I fear for the welfare of my sons should they return to her native country, or even if they should stay in the USA, in her custody, as part of a divorce.
She is intending to leave me for no reason other than her own selfishness, and has no concern for our sons growing up without a father.
Quite frankly, I am a far better parent than her.
Since she has lied to Homeland Security about her previous marriage, and now wants to leave the country anyway, I see her being deported as a much cleaner - and less financially and emotionally devastating - option than a messy, expensive divorce.
I'm 64 (She's 35), and don't want to be without my sons, and paying child support until I'm 81 (Guiness record?)
Any advice?

A very familiar story. I assume she has a permanent resident card as opposed to the two year temporary resident. If she still has the two year, deportation would be much easier. However, if she has the permanent resident card (10 year) it is more difficult to deport her. I am not giving legal advice here, just passing on what I have read as I am going through a similar situation. If she has any criminal record, that is where you want to start. Even a crime such as shoplifting carries a major penalty as it is defined by immigration as "moral turpitude". As I understand it, one of these violations in her first five years of being in the states with a possible sentence of one year (even though she may not actually receive that sentence) wins her a free boat ride back to China (or wherever she's from). But it is possible to get a waiver of deportation from the judge for this if she has "extenuating circumstances. But more than one conviction, regardless of potential sentence or how long she's been in the country also guarantees deportation. Waivers aren't possible with this method.
Now the bad news. My wife has four shoplifting charges. She had two when she applied for citizenship and was denied (but not deported). She now has two additional shoplifting charges which should be more than enough to get her deported. But it seems immigration is only concerned with illegals rather than those who break the law constantly while on a permanent resident card. Oh yeah, she also lied to get a Pell Grant (she already has a BA and MA from Mongolia) and the government isn't doing anything about this either.
Conclusion, if she isn't breaking the law, you're going to have a real hard time deporting her. As you can see, even if she breaks the law on numerous occasions the government doesn't seem to want to use their time unless it is felonies despite their own rules and the LAW.

You are not the victim here. Everyone who marries a foreign national knows that US citizenship is often a factor in the relationship, especially with a 30 year difference in age. If she leave the country with your children you will be in a no win situation. You may be the better parent but you are also 64 and she is 35. Age will play a factor in custody. You have a domestic problem. Involving immigration will only make things worse. I would try to make a living arrangement that makes it a positive environment for her to want to live near you so your children can be in your life. The children should be your objective. Having a relationship that works for both of you is in your best interest. The alternative is a winner take all approach. As a father who lost my child in a transcontinent feud I can tell you any amount of children in your life is better than 100% of nothing.

Why you want her to be deported? ...

And if she intends to leave anyway, what would be the point?

It is doubtful that she can get U.S. passports for the children without your permission. Whether she can get them passports from her home country is another matter; I guess it depends on the law of that country.

Deporting her or not is the government's business, not your business. Your business is limited to looking out for your interests and the interests of your children. And with respect to that, her immigration status should not be relevant.

Custody of the children will be decided by a court. If you have good evidence that she is not good for the children, you can present that. The court will hopefully decide fairly, but what you have said so far doesn't seem to be very convincing evidence.

If she has full custody of the children, she can legitimately bring them wherever she wants. Otherwise, it will need both of your permission to get a passport (for most countries). If the children already have passports, or she can get passports for them for their other nationality and that country doesn't require your permission, then it may be possible she may be able to bring the children away anyway. If that is the case, you can report a kidnapping.

If she wants to divorce, then you guys divorce. This is America. People divorce all the time. If she is not going to be together with you, then divorce as soon as possible is in the interest of both of you. After you divorce, whether she leaves the U.S. or stays or whatever, and how she stays, is her problem. I don't see much that affects you. I would suggest you stay out of it; retribution is not a good road to go down.

With respect to her status, I believe that having been married for several years, with children born into the marriage, she would not have a hard time proving that the marriage was originally bona fide.