I can't speak to the Roman issues, but a big part of what you're talking about here has nothing to do with an age of majority (which is not inherently the same thing as an age of consent) varying.

In medieval Europe, it was very common for royal children to be betrothed while still quite young, allowing their parents to use them as bargaining chips in peace treaties and alliances. Their youth didn't really matter because they weren't going to get engaged until they reached adolescence; in a number of cases, they didn't get married at all because later events caused the engagement to be called off on their behalf. Mary Tudor (the future Mary I) was betrothed at the age of 2 to the dauphin of France during a cessation of hostilities between France and England, but it was broken off when she was 5; a year later, she was engaged to her cousin, Charles V, the Holy Roman Emperor, in order to make a Habsburg-Tudor alliance against the French. This second betrothal was broken off by Charles so that he could get married to a woman who was already an adult and could give him heirs soon. It's not that there was no expectation of these infant engagements to come to a wedding, but it was understood that they were more about showing agreement than a definite promise to get married. It was less common for a marriage to actually go through between a child and adult, like the six-year-old Isabella of Valois marrying the adult Richard II of England and going to his court. In that case, the need for alliance was strong enough that something more than a betrothal was needed to cement it. However, marriages with children weren't fully valid until the child reached the age of consent (not majority) and could affirm or revoke the bond; if she affirmed the marriage, then it could be consummated, although it was safer to wait a bit beyond the age of consent to make sure that she was physically capable of bearing a child.

What happened with Margaret Beaufort is a little different. Her father, the Duke of Somerset, died when she was very young, and although her mother was still alive and kept custody of her, that meant that Henry VI was her guardian and had the right to assign her to be the ward of some other powerful man. Wardships of heiresses (Margaret was the duke's only legitimate child) were highly sought-after because it was very easy to force a marriage with the man himself or his son, if he already had one, and thereby get her money and land into his family. Normally the guardian would wait until the ward's adulthood, but in a situation where one really wanted to make sure, the wedding could happen between children - it would be unconsummated, and could be dissolved when the child bride reached the age of consent, as described above. The duke had been arrested and was on his way out, making him desperate. But the king dissolved the marriage himself, making the point moot, and gave her as a ward to his half-brothers, Jasper and Edmund Tudor - while also arranging for her to marry Edmund when she was twelve (the age of consent for girls) in order to give him some more legitimacy as a potential heir, as she was a legitimate descendant of Edward III. Again, desperation! And desperation from the political unrest of the onset of civil war led him to consummate the marriage immediately to provide a future heir for himself, potentially at the expense of her own life. We can't know that he wasn't a pedophile, but there were at least circumstances that made it "logical" for her marriage to turn physical so early. She was valuable as a maker of heirs and as a conduit for her wealth, land, and bloodline to be transmitted; her physical well-being and her mental state were not important to the Tudor family.