Short answers, pulled from the story.
A heraldic badge is a graphic symbol worn or displayed to show allegiance to, or the property of, an individual, family, or corporate body. Unlike a coat of arms, which was used exclusively by the individual to whom it was granted, badges are para-heraldic and their use is more flexible; they could be worn by retainers, household servants, and followers, not just the owner.
Parliament repeatedly tried to curb livery badges because they were used to mark the private armies of retainers kept by lords to enforce their will on the less powerful. In 1388, Parliament requested that all badges be abolished, citing the extortion and arrogance of those who wore them. The Statute of Liveries of 1506 finally forbade issuing liveries to those of higher rank unless they were domestic servants, persons experienced in the law, or covered by a royal licence.
The Dunstable Swan Jewel is a rare surviving example of a lavish medieval badge made in enamelled gold. A badge of that quality would only have been worn by the person whose device it represented, members of his family or important supporters, and possibly servants in regular very close contact with him.
In 1483, Richard III ordered 13,000 badges in fustian cloth bearing his emblem of a white boar for the investiture of his son Edward as Prince of Wales. Other surviving grades of boar badges include examples in lead, silver, and gilded copper relief; a gilded copper example was found at Richard's home of Middleham Castle in Yorkshire.
The Tudor rose was adopted by Henry VII of England to signify the union of the Lancastrian and Yorkist dynasties, the two warring houses of the Wars of the Roses. It combines the White Rose of York and the Red Rose of Lancaster, and was used more widely than earlier royal badges, including being added freely to King's College Chapel, Cambridge.
Heraldic badges were formally revived in 1906 by the College of Arms under Alfred Scott-Gatty. Since then, badges have often been included in new grants of arms; the grantee pays a higher fee to receive one, and a heraldic standard bearing the badge may also be requested for an additional fee.