mick,
Short answers:-
1) Depends.
2) Sort of.
3) Yes.
4) Probably.
Long answers:-
1) All British troops in North America (which at that time included the Caribbean) would have had two sets of clothing at the start of the war. Their "best" uniform was the latest clothing issue for that year - assuming it had reached them! This would generally be kept back and worn for the first time at the King's birthday parade on 25 June; it would then be reserved for parades and (if they were not operating at a distance from base) in battle. This would conform reasonably closely to the 1768 Clothing Warrant. And forget "Carry On Sergeant" and the National Service horror stories - each man's clothing would be carefully altered and fitted by the regiment's tailors. For everyday wear, the men would cut down the previous year's clothing into "slops" or what we would call fatigues. This would be tailored to the local climate, but broadly would consist of either a "frock" (single breasted) coat and/or sleeved waistcoat.
In the first year of the war, whether units went into battle wearing "best" or "old" depended on the whim of the CO, how far they were operating from their base, and what type of mission was originally anticipated. For example, the Lexington/Concord raid was only expected to be a search for, and destruction of, illegal military stores; so the flank companies quite possibly wore hats instead of their bearskin and leather caps, given that none of the latter were reported lost. On the other hand, we know that the grenadier company of the 14th Foot donned their bearskin caps for the fateful dash across the causeway at Great Bridge in Virginia because some such items WERE reported as lost.
There are contemporary illustrations of soldiers in Boston wearing cut down hats and coats in 1775, but they are undertaking everyday duties. Whether this clothing was worn in action, is open to debate - IMO they probably wore "proper" uniforms at Bunker Hill, as fighting was expected. In August 1776, a German colonel referred to the British wearing trousers or overalls in the battle on Long Island, whilst the Hessian troops were still in their long black gaiters. The Foot Guards actually arrived in mid-1776 wearing cut down slouch hats and shortened coats that had been modified before they left England. Also, Howe was a light infantry commander in the French & Indian Wars and so had strong views on how troops were dressed on campaign in America. So the long coats and tricornes look is by no means certain after Bunker Hill.
2) Burgoyne's regiments cut down their old coats and hats - the ones they were wearing when they arrived in early 1776 (ie their 1775 clothing) - to a specific design; the spare material was used to patch and mend. This was done because several regiments had their new (1776) clothing captured by enemy privateers en route to Canada (those who did not, were still required to make the amendments in order to create a sense of uniformity). [Note that the campaign was due to start well before the 1777 issue arrived.] Howe's army at New York mostly received its 1776 clothing; its modifications were mostly aimed at making the uniforms more "user friendly" in the field. Apart from the Light Infantry battalions - who adopted the "slouch" hat and "roundabout" waistcoat - what modifications were carried out were generally at the whim of the CO of each unit, and we don't even know for sure that all of them made any as not all of the orderly books have survived. There was no standard "campaign dress" for Howe's army.
3) The "slouch" hat was worn in all theatres and was probably fairly universal in the field after 1776 as it prevented the hat being knocked off by the musket when marching, but also provided shade to the eyes and neck (it gets very hot in summer, even in New England and the mid-Atlantic colonies). In the British forces, the style is particularly associated with the Light Infantry. Essentially, all that happened was that the hats arrived completely uncocked (ie "round" hats) and were just cocked on one side, according to a pattern that was usually held at the quarters of a senior officer and could be viewed by someone from each battalion. A common style in the South was to turn the cocked side to the back in order to give a substantial shade to the eyes.
4) Roundabouts - referred to at the time as "postillon" or "Newmarket" jackets - were the ordinary soldier's waistcoat, usually with the sleeves of the regimental coat sewn on. The term usually refers to the red waistcoats worn by the Light Infantry battalions. It is less likely that such an item was worn by the grenadiers or centre companies, since their waistcoats were white. That said, some units may have cut down their old coats to make single-breasted items and these could have been seen in any theatre; the Light Infantry version was probably worn in all theatres where they served.
On a general point, I don't know why, but re-enactors get very cross if you call it a tricorne - I think possibly because it's a civilian term and the Army called it a cocked hat. Personally, I think it's a good way of indicating to people not familiar with the period that a hat is cocked on all three sides, as opposed to one side (slouch) or none (floppy).