The "Mad Minute" has been greatly hyped-up. Many well trained soldiers were able to attain 15 rounds per minute, but most were able to achieve 8-10 rounds per minute. Without needing to reload thanks to the magazine of their Lee-Enfield riddle, this did give the regulars a huge advantage.

Quality declined for two reasons: 1) the terrible losses the regulars suffered in 1914; and related to #1, 2) the arrival of the Territorials and Kitchener Volunteers. Most of the original rank and file of the BEF was dead, wounded missing or captured by December 1914, the BEF having seen fierce fighting at Mons, Le Cateau, the Marne, the Aisne, and the bloody 1st Battle of Ypres. This meant that the ranks were increasingly filled with Territorials (reservists) and the 1914-15 'Kitchener Volunteers'. The de-skilling of the BEF contributed to it's teething pains and difficult performance in 1915 and 1916, but from the Battle of the Somme the BEF was transformed into the effective fighting force that would spearhead the Hundred Days Offensive in 1918.

In addition to what /u/DuxBelisarius said, it's important to remember one the key difference between the British Regulars and most of the land armies of Continental Europe is that they were a purely volunteer based army, whereas the continental nations, including Germany, France, Russia, and Austria, all used conscription. This means that Britain had a small but highly professionalized set of soldiers, while other countries had conscripts that eventually phased out when they finished their required service.

Since Britain was an island and a sea power, they focused largely on their navy, and focused on making their army a small but very well trained and highly professional land force that would largely serve as a way to police its sprawling possessions. Therefore, Britain's volunteer army were trained quite well and became one of the best armies in Europe in terms of quality, despite being grossly outnumbered by most Continental armies.

British Regulars did end up being some of the best soldiers in the war throughout 1914-1915. They were ferocious and very skilled, and contributed quite heavily to Germany's eventually slow down on the Western Front in 1914. They massacred waves of Germans in the First of Ypres (though many, not all, but many of their German counterparts were inexperienced students), just to name a battle. In many battles they held their own and inflicted higher casualties on the Germans, but were forced to withdraw nonetheless simply because the Germans had more men and overpowered them, or because their flanks were being collapsed by either other British, French, or Belgian troops buckling.

They were largely wiped out by mid 1915, 1916, however, simply due to the sheer amount of casualties suffered on the Western Front. Germany's army in the second half of WW1 was almost completely different from the army it started the war with, with most of the original soldiers either dead, wounded, missing, or taken prisoner. Britain suffered a similar fate and many Regulars had to replaced with Territorials, Volunteers, or Dominion troops from Canada, ANZAC, India, or other places in the British Empire.

Generally speaking, yes. They had been schooled in the realities of modern war by the Boer War, and emphasis on firepower had become a key doctrine. Cavalry and infantry alike were armed with SMLEs and were well trained in their use. German officers reported that their men were "shot flat" by British musketry at ranges that they, the Germans, could not respond to.