In a world of bastards there is no shame in being a bastard.

And today's world is choc-a-bloc with them.

They weren't considered bastards even in medieval times, when they were far more strict about that sort of thing and the issue of whether or not a child was a bastard was even more important.

Perhaps you can support this with a source. I could imagine that given how rare and unusual annulments were, that any children involved might not have been considered bastards, but it's purely speculation on my part until a source can be provided.
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On a related note, when a couple living in sin (not married in the Church, for example) has their relationship validated in the proper manner, does that alter the status of any children they bore previously?

I don't remember. Possibly something he said in the social area, like referring to a stepchild.

This thread is about annulments, which has nothing to do with marrying a divorcee. The Catholic Church has never granted divorces.

On a related note, when a couple living in sin (not married in the Church, for example) has their relationship validated in the proper manner, does that alter the status of any children they bore previously?

In the 1983 Code Canon 1139, illegitimate children are legitimized by a later marriage of their parents (valid or putative). A putative marriage is one attempted within the Church, but invalid.

In the 1917 Code Canon 1116, they are legitimized by a later marriage (again, valid or putative) if the parents were capable of contracting marriage at some point between the conception and birth of the child.

Both codes say the the children conceived or born in valid or putative marriages are legitimate. The 1917 Code Canon 1113 has an exception for marriages with a person in solemn religious profession or holy orders.

Perhaps you can support this with a source. I could imagine that given how rare and unusual annulments were, that any children involved might not have been considered bastards, but it's purely speculation on my part until a source can be provided.
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On a related note, when a couple living in sin (not married in the Church, for example) has their relationship validated in the proper manner, does that alter the status of any children they bore previously?

Most annulled marriages are still considered to be putative marriages, i.e a marriages where at least one party entered it into good faith believing it to be valid. That children borne of putative marriages are valid has been part of Canon Law for a long time. I can't find any direct sources on showing this concept/law to have been present in Medieval times, but others have already sourced it in Canon Law dating back long before Vatican 2. As for a secondary source regarding the Law in Medieval times, Harold Berman's "Law and Revolution, the Formation of the Western Legal Tradition" describes it to have been the case back then too:

p228: "Where the parties married in good faith, without knowledge of an impediment, the canonists held that the children of the marriage were legitimate and that the marriage itself was valid up to the day it was declared null."

I don't have the book on hand to look at his primary sources for this, but the fact that we know it to be true in 1917 Canon Law means I'm inclined to believe him that it was the same back then too.

Interesting: do you endorse rash judgment about grave sin?

It's not rash judgement.  Your Uncle married a divorced woman.  Why, or how, is it known that she was a divorced woman?  What is the level of Catholic knowledge or commitment in a man who does such a thing knowingly? 
Rash judgement would be saying that your Uncle is on the way to Hell.  That would be rash. 
Advising him to seek out a Traditional Catholic Priest for advice on his current marriage would be a prudent thing to do.

It's not rash judgement.  Your Uncle married a divorced woman.  Why, or how, is it known that she was a divorced woman?  What is the level of Catholic knowledge or commitment in a man who does such a thing knowingly?  
Rash judgement would be saying that your Uncle is on the way to Hell.  That would be rash.
Advising him to seek out a Traditional Catholic Priest for advice on his current marriage would be a prudent thing to do.

1. While my uncle did marry a divorced woman, that doesn't mean he wasn't validly married. Could be any number of reasons, including: Pauline/Petrine privilege, annulment of her previous marriage, or the previous spouse is now deceased. Terminology can be muddied in the modern world. Yes, sometimes people continue to identify as divorced even though they are not, or a widow, by Catholic thinking.

2. Many circuмstances can affect a marriage. Especially if prior attempted marriages are involved, things can be complicated. Then there are issues of justice for any children. I do know what was going on with my uncle. The busybody who approached me about my uncle knew who I am and what my background is. The person frankly should have trusted that my uncle's situation was already checked out. Should I have had to explain my uncle's marriage to everyone who asks? There can be circuмstances of a marriage case that are "private" or inappropriate to discuss, perhaps because they could reveal or imply sins of another, and so would be an injustice.

3. I'm saddened to see the people arguing about this. My post referred to a busybody who didn't have all the facts. Arguing against me has been a resistance Mass coordinator who went to SSPX seminary but (if I recall correctly) never studied moral or pastoral theology there. The one who accused me of being "Bergoglian" also went to SSPX seminary but (if I recall correctly) also never studied moral or pastoral theology there. I don't know you, but still that's three people doing pretty much what I was identifying with the busybody. None of you know the circuмstances but still seek to provide "advice". This probably falls short of"rash judgment" due to lack of firm judgment (perhaps it's "rash suspicion"?). Nevertheless, this sort of behaviour is something I associate with the old ladies of a parish.

4. More broadly, this thread is an example of a trend that gives me pause about the resistance segment of tradition. I agreed the way annulments are handled in the US isn't right. But here I get "downvotes" for expressing some traditional pastoral theology, and some leading lights of this forum appear to argue against that theology, based on an ideal of what society should be but currently isn't, or because it sounds liberal. This is an example of why I suspect even when/if Rome did return to tradition, it may not be accepted by many traditionalists, who sometimes have peculiar notions of "tradition".

You were the one that brought up the conversation, which like I said, has nothing to do with the OP, Annulments. Your posting derailed this important thread. Now you are derailing again by disparaging the so-called Resistance people.

OK, to be on topic: you mentioned advising people who have had annulments. And while I'll agree it's not directly relevant to annulments, my uncle's case does relate to advising on marriage status.

Do you recognize the potential problems with confronting people on their possible invalid marriage or "living in sin", when they may very well be doing nothing wrong? When they may have already had priests look into the situation? 

You also said:

People who do not know a subject, are not in a position to advise anyone, that should be obvious.

Most laypersons do not have the background in moral theology to advise on marriage cases.

Also, the people who would be advised often don't have enough background to explain their situation accurately. They may either omit relevant facts, or use terms that can be misinterpreted.

Prudence should dictate care and caution in this realm, and men should have enough self-control not to jump at suspicions.