"Officer" is a bit too big of a category to answer this across the board, as some of these conditions changed based on duty, and they all changed on rank. Equipment also changed throughout the war, sometimes pretty drastically.

The first thing to remember is that officers were responsible for their own clothing and equipment. Their uniforms were built to the same color and pattern as the rest of their regiment (or at least, what that regiment was supposed to be wearing), but could vary in terms of material. There really was no standard pattern for swords or spontoons, and most officers were not supposed to carry pistols (more on that later). Officers tried to make their uniforms out of better material than the enlisted men's coats, both for comfort - good wool is lighter - and because maintaining and broadcast one's fine appearance was a way of reinforcing the social hierarchy that helped keep the enlisted in line.

Let's start with a line infantry officer, post-1779. I chose this year because the army has had, at this point, at least two years to seriously professionalize on the von Steuben model, and over a year for the flood of French supplies to disseminate through the army. His coat would be blue wool, with facings color-coded by region - white for New England, red for the middle states, and lighter blue for the South. Per the general orders of December 22, 1777, officers were not to carry firearms of any type, as "firearms when made use of with drawing their attention too much from the men." Basically, if officers were futzing with pistols or fusils during combat, they would be keeping their eyes on the firings and steadiness of their commands. Instead of a firearm, officers were supposed to carry a spontoon, a roughly 8-foot long spear. Officers were also supposed to have swords, either purchased from one of the fairly small number of American sword makers, captured from the British, or a gift/purchase from the French. His short clothes - those worn beneath the regimental coat, were either of linen or wool, depending on the season and the ever-present vagaries of supply.

These supply problems meant that even officers were occasionally ragged in appearance, though this was heavily discouraged. An artillery officer was called out in the general orders during the winter of 1780-81 for being too shabbily dressed to command respect. Still, given the abysmal state of the American economy, there was only so much one could do. Still, officers, ever class-conscious, tried every possible means to make sure they carried themselves and professionals and gentlemen.

Feeding a large number of horses was a major problem for both armies during the War of Independence, so the number of officers riding horses was relatively low. In the infantry, only majors and above were supposed to have horses, and (I think) only generals were allotted spare mounts. Again, these horses were provided at the officers own expense, and were not property of the army. Washington kept two horses, Nelson and Blueskin, with him throughout the war.

Outside of the infantry, these answers changed. Lieutenant (and future governor of New Jersey) William S. Pennington of the 2nd Continental Regiment made several mentions of riding a horse, though it is unclear (to me, at least) if this was his normal behavior of if he had special duties within the regiment. Cavalry and some dragoon officers, of course, rode horses, and were also authorized to carry pistols.

Pistols are an interesting point to get into. As mentioned above, officers were not supposed to carry them into battle or on duty. And yet we know some officers, including Washington, owned them. These seemed to have been used primarily for dueling - which meant, of course, that they were not used all that frequently, but it was important that a gentleman be ready to defend his honor. Would these men have had the discipline to own these weapons and not carry them into battle? That's hard to say, and I don't think the sources provide an answer. Pistols were pretty useless outside of a duel, anyway, due to their short range and terrible accuracy. Certainly, an officer who found himself within firing distance of the enemy would be better served by directing his soldiers deal with the threat than by doing so with his own weapons.

Sources Carolyn Cox, A Proper Sense of Honor

John Clement Fitzpatrick, The Writings of George Washington

Jim Kochan and Don Troiani, Don Troiani's Soldiers of the American Revolution

The Diary of William S. Pennington