1. Although they are retrospectively applied to earlier Monarchs, medieval European Monarchs did not actually use ordinals at their own time. The English tradition of numbering Monarchs dates back to the reign of Henry VIII, although some attempts at numbering the Monarchs had been made as early as the reign of Edward III.
2. There isn't a single person who just decided to use ordinals; rather, they appeared with time as a convenient method of distinguishing Monarchs who shared the same name. Popes were probably the first to assume official ordinals for their reigns.
3. The kings of England are numbered starting with the Norman Conquest for convenience. Although starting with Egbert, most Kings of Wessex chose the title "King of the Anglo-Saxons", and starting from Athelstan - "Rex Anglorum", none were actually Kings of ALL of England. Separate, smaller Kingdoms continued their existence, although most were directly under the control of the King of Wessex. William the Conqueror was arguably the first King to actually have control over all of England. That is why the son of Henry III of England is counted as Edward I, even though there were three Edwards before the Conquest (Edward the Elder, Edward the Martyr and Edward the Confessor).
4. The name of the country did not change in 1603. When James VI of Scotland succeeded Elizabeth I as James I of England, that did NOT mean England and Scotland were united; rather, the two countries enjoyed Personal Union of Crowns - that is, shared the same Monarch while remaining separate, sovereign Kingdoms (similar to Canada, UK and other countries of the Realm now). The countries were only united with the Act of Union 1707.
Most of the Monarchs after James and before Act of Union had names that had never been used before in either countries. Those who didn't, continued to have separate ordinals for each country.
- James was James VI of Scotland and James I in England.
- Charles I was I in both countries (neither had a King Charles before)
- Charles II was II in both countries or the same reasons as his father.
- James II was James VII in Scotland and James II in England.
- Mary II was II in both countries because both England and Scotland had a previous Monarch with the name (Mary I in England and Mary, Queen of Scots in Scotland).
- William III (Mary's husband and co-regnant Monarch) was William III in England and II in Scotland.
- Anne was Anne I (though the ordinal wasn't used) in both countries because neither had a previous Monarch with the name.
With the Act of Union 1707, separate ordinals were no longer needed because one single country was formed. No issues arose with the first five Monarchs after the Act of Union 1707 (Queen Anne, George I, George II, George III and George IV) because neither country had previous Monarchs with the name. However, William IV (who should have strictly speaking have been William III in Scotland), it was informally decided that the subsequent Monarchs would use the highest Monarchial ordinals of England.
The issue with numerals arose again when Queen Elizabeth ascended to the Throne; the previous Monarch with the name (Elizabeth I) was Queen of England only, while Scotland has never had a Queen of that name. To avoid any future controversies, Winston Church suggested a rule that all future Monarchs of the United Kingdom and the Commonwealth Realms will use the higher of the two Monarchical ordinals of England and Scotland. Thus, a future King Henry would be Henry IX (although Scotland has never had eight Kings named Henry), while a future Queen names Margaret would be Margaret II (because Scotland has had a previous Queen Regnant names Margaret - Margaret, Fair Maid of Norway). The Commonwealth Realms also agreed to this suggestion, which is why Queen Elizabeth is Elizabeth II in Canada, Australia, New Zealand, Tuvalu and other Realms, though none of those countries has had a previous Monarch named Elizabeth.
5. The name of the country has indeed changed several times since 1707; first, it was Great Britain, then United Kingdom of Great Britain and Ireland, then United Kingdom of Great Britain and Northern Ireland. However, the ordinals didn't change because each was a successor state of the other, with all legal consequences. England/Britain/UK is not unique in that respect; nearly all Kingdoms of the past and present have gone through similar changes and transformations throughout their histories.