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How does being a convert enter into it?

I was trying to highlight the case of a catholic marrying a non-Catholic outside of the Church.  In that case, there is no sacramental marriage, though there is a natural law marriage.  If the non-Catholic converts, then they would be obligated to get it blessed by the Church.
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If you have 2 non-catholics get a natural marriage, but one converts to the Faith, while the other is anti-Catholic and harsh towards the Faith, the convert could get "re-married" to a catholic.  It's called the Pauline Privilege, as it apparently was supported by St Paul, during the early days.  My explanation is simplistic, but this is what I've been told.

Many, or most, cases today would fall under the conditions of Canon 1098 of the 1917 Code, when the authorized pastors and ordinaries, or a delegate of either, prescribed by Canon's 1095 and 1096 cannot be had without great inconvenience, and it may prudently be foreseen that this will last for a month.   As long as there are no diriment impediments, all that matters for a marriage to be valid under the conditions of c. 1098 is the fact that the conditions of c.1098 truly exist when and where the marriage takes place.  Under such conditions, mixed marriages between a baptized Catholic and a baptized non-Catholic are valid, even if in bad faith they are married by a civil official or non-Catholic minister in a non-Catholic ceremony.

Isn't a mixed marriage only considered "valid" as a "natural law" marriage?  It surely isn't a sacramental marriage, so it's not valid, from a sacramental/Church aspect, but only valid from a human promise aspect.  This would all be true even if the couple was married by a priest, no?  He can bless the marriage, but it's still not sacramental.  If the couple didn't get Church approval, it would be a sin.  If the priest performed the ceremony without approval, it would also be a sin.
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Certainly if any catholic gets "married" by a non-Catholic (in any shape), that is not a catholic/sacramental marriage.  In fact, such an act would be a grave sin.  It would not be a marriage at all, neither natural law or sacramental.  The best the couple could hope for, spiritually speaking, would be if they stayed together and God blessed them as fulfilling their natural vows.  But spiritually speaking, this would be the worst-case scenario for a catholic, only slightly better than co-habitating (but maybe, from a moral theology standpoint, the same grave sin as co-habitating).

These must be proven and verified, but I was taught they are all pre-Vatican 2 annulment grounds.
Perhaps Pax can corroborate.

I'm not an expert on this area at all, but it would stand to reason that if a person lied about a grave matter/circuмstance, that it would be grounds for an annulment.  Mental issues, extreme violence, alcoholism/drug addiction...all these would make one question the ability of the person to fulfill the marriage duties, so hiding such personality defects would be a potential contractual impairment.