Having just recently attended a "shotgun wedding" myself, just curious in the Medieval Ages did it matter for inheritance for royals or high nobility if it was clear the precious little heir-to-be was obviously conceived before the wedding ceremony?

That's an excellent question!

The medieval Catholic Church often gets a bad rap on matters of marriage and sexuality, but the historical record often shows them to be more moderate and reasonable (if still profoundly premodern and often quite weird) than their reputation would suggest.

Specifically on the topic of bastardy, the medieval Catholic Church was not nearly as harsh and unforgiving in its canon law as the Westerosi. The Church's attitude was that, while sex out of wedlock was a sin, it was a sin that belonged to the parents and not to the child, so punishing the child would be unjust.

So what about the in-between area of a bride-to-be with a premarital pregnancy? Well, as far as canon law is concerned, a true shotgun wedding wouldn't be valid because the consent of the people getting married has been coerced - and to their credit, the medieval Catholic Church really cared about consent in marriage to the extent possible in a patriarchal society where marriage alliances were absolutely central to the acquisition and transfer of landed property and political power.

However, if the couple was just merely enthusiastic about their engagement and genuinely consented to be married, premarital pregnancy is not considered a canonical impediment to marriage and the marriage would be considered valid.

Likewise, because the Catholic Church saw it as its mission to encourage marriage and birth within wedlock and that punishing a child for the sins of the father was unjust, the Church was (and is) pretty liberal in how it treats the legitimacy of children. So for example, a child born but not concieved in wedlock is considered legitimate in canon law and thus would be entitled to baptism, confirmation, marriage, and other sacraments and would be treated like any other legitimate child in so far as their inheritance rights were concerned.

Does the case of Fireball's marriage have a historical equivalent in the Middle Ages or Early modern period? If a woman took the holy orders, was her marriage dissolved and thus her former husband could make a new canonically valid marriage? Was this also the case for the wife if her husband took the holy orders? Was it specific to monastic vows, ordained priests (after celibacy became the norm) or other specific clerical statuses?

Excellent question!

When it comes to the case of a husband taking Holy Orders, this was a matter of some contention within the medieval Catholic Church and quite a few big name Popes exerted a lot of political capital to make clerical marriage stop. (Partly this was due to theological reasons, but mostly it came down to not wanting married priests to try to convert Church property into something their children could inherit.) So it depends on when in the Middle Ages we're talking about; the earlier you go, the more likely you are to have married priests, the later you go, the less likely that becomes. And then that wascally Martin Luther fell in love with Katharina von Bora...

The issue of women joining nunneries was less politically controversial, because it didn't raise the issue of Church property being converted from corporate to personal property. Indeed, the Church had something of a interesting incentive in the matter, because women who wanted to become nuns had to bring a "convent dowry" with them, and in the case of wealthy women or heiresses, these dowries might be rather substantial.

However, when it came to either husbands or wives entering Holy Orders, this was considered to dissolve a marriage...with one main condition. In the words of Pope Alexander III, "it is permitted for one spouse, even against the other's will, to choose the monastery...so long as a joining of the flesh has not yet occured between them."

So that could be a real sticking point. I suppose a man in Fireball's situation would need to find an abess who was willing to accept that the marriage had never been conssumated for a hefty bribe.