Short answers, pulled from the story.
The Tudor court employed painters from across Europe, most prominently Hans Holbein the Younger, Lucas Horenbout, Levina Teerlinc, and Nicholas Hilliard. Many were Netherlandish, though artists also came from Germany, Italy, and France. Some English artists, including George Gower and John Bettes the Elder, also worked at court.
John Browne was appointed the first Serjeant Painter in 1527, having served as heraldic painter since 1502 and as King's Painter from 1511-12. The post was created when Lucas Horenbout was brought in as the new King's Painter, making that the superior position. Browne died in office in December 1532.
Nicholas Hilliard served as miniaturist and goldsmith to Elizabeth I from around 1572, and is considered the supreme miniaturist of the Tudor period. He received a gift of 400 pounds in 1591 and an annuity of 40 pounds from 1599, and typically charged 3 pounds for a non-royal miniature. He also taught the art of limning to Isaac Oliver and trained other apprentices including John Bettes the Younger and Rowland Lockey.
The Reformation virtually eliminated religious painting in England, making portraiture the dominant form for all Tudor court artists. It also disrupted artistic contacts with the rest of Europe, especially Italy. However, England also benefited from the Reformation's upheavals, as Protestant refugees from Flanders - including Marcus Gheeraerts the Elder, Marcus Gheeraerts the Younger, Lucas de Heere, and John de Critz the Elder - brought skilled artists to the English court.
Nonsuch Palace was begun by Henry VIII in 1538 south of London and was intended to compete with, and outshine, the Palace of Fontainebleau. It was covered inside and out with figurative sculpted stucco reliefs spanning more than 2,000 square metres. Italian artists were brought in from Fontainebleau for the Mannerist work, and Antony Toto and Bartolommeo Penni spent most of their time after 1538 working on the palace. The palace has since vanished.
Many Tudor portraits have been badly worn or incompetently restored over the centuries, and inscriptions are often later additions that may reflect wishful thinking rather than genuine knowledge. Artists' workshops produced copies of portraits on demand, and not all artists signed their work consistently. A portrait identified in 1727 as Lady Frances Brandon was only correctly reattributed in 1986 to Mary Nevill, Baroness Dacre, with Hans Eworth identified as the artist.