After the war they implemented a democracy with a limited government

Correction: They implemented a republic, not a democracy. These are not interchangeable terms; they mean different things. The Founding Fathers, almost without exception, despised the idea of a democracy in the traditional sense (i.e. direct political involvement by every eligible male citizen, bypassing the need of political representation). They viewed democracy as chaotic, unstable and too prone to radical populism, which they considered as being the downfall of the Ancient Greek democracies. The Founding Fathers were inspired, rather, by the early Roman Republic, with its various and multi-layered assemblies whose members were politicians voted into power by the citizens, and would then in turn elect the executive leader themselves (consul for the Romans, president for the Americans).

The Revolutionary War's initial grumblings begin, at least in my current understanding, with the Royal Proclamation of 1760 dictating that the colonists stop going around stealing land from and pissing off the natives, up until then nigh-infinite land for everyone who wants it being an incredibly important aspect of the American dream compared to Europe where all farmland had been neatly partitioned off for generations.

This is correct. A major source of tension between several of the colonies (most notably Virginia, the most ambitiously expansionist one) and Britain was over the right to expand west into lands that had, for the most part, been recognised by the British government as rightfully belonging to the Native Americans. The British government was eager to keep as many natives on their side as possible to counteract any French ambitions in the region (Native Americans had fought on both sides in the Seven Years War), and also didn't want to have to station too many soldiers to protect the colonists in the frontier regions from native attacks, as they viewed this as an unnecessary expenditure. In the eyes of the British politicians, further expansion west wasn't necessary - the Thirteen Colonies were already enormous by the standards of European countries, and with the recent acquisition of most of Canada in the Seven Years War, they already had a new vast American territory to settle. But ambitious settlers - mostly Virginian - eager to expand their land holdings weren't content with these arguments.

After this the taxes sought by the British government were a fair asking point from the colonists after the French and Indian war had devastated the British coffers (genuine point of ignorance on my part here, was that a good thing for the colonists or was it just Britain maintaining the right to profit off of colonies that would be fine either way?)

The origins of the war are a somewhat complicated topic, and the blame isn't squarely on either party. The French and Indian War (or Seven Years War, as it's known elsewhere) was started when the French violated a treaty signed with Britain by building a military fort on a river in an area where they weren't permitted to. The British viewed this as French attempts to control the commercial traffic of the river (whose name escapes me at the moment), and the British had vested interests in keeping the river a "neutral" zone for free trade (a free trade which was, in reality, dominated by the British, whose cheaper goods were more attractive to both the colonists and natives alike). When the French refused to pull down the fort, the British initiated hostilities in a small skirmish led by George Washington. The French retaliated by invading the north-west area of the Thirteen Colonies. As you can see, both parties were responsible for the situation escalating to war, but once the French invaded the Thirteen Colonies, the American settlers were most definitely involved, and Britain ended up sending thousands of troops over from Britain to aid them in the war effort (the colonial militias had proved to be fairly weak forces).

Ultimately the war started over the neutral status of a river that was not owned by the British but was highly valuable because it provided easy movement of cheap and popular British goods in North America. It was a war started for free trade; a free trade in which the British were dominating because they were ultimately better businessmen. While the French had no intentions of conquering the Thirteen Colonies (with a population of 90,000 compared to the British settlers' 1.7 million, this was never going to happen), they did invade several times, with varying success, necessitating British defense of the colonies. And thus lies the argument for compensation. However, while the Thirteen Colonies certainly enjoyed the benefits of these cheap British goods, they weren't the ones that decided to go to war over it; that war was brought onto them by decisions made by British and French policymakers in the Old World. And that's the counter-argument by the colonists; their lack of political representation dragged them into a war they didn't vote for, and couldn't vote for.

After the war the founding fathers created a fairly bougie system where in landowning gentry were the sole politically empowered group in this country. Overall it seems like that patriots, or at least their leadership, were a bunch of rich guys just looking out for their own bottom line and fighting a war to protect it which is something I condemn entirely during the later American Civil War.

You're correct in that the Founding Fathers were almost exclusively belonging to the upper class of colonial society, and established a system of limited suffrage, but it would be too reductive to simplify everything to "rich people not wanting to pay taxes". As the Founding Fathers outlined several times during the Continental Congresses, they weren't inherently against the idea of taxation - as a bunch of lawyers, they were realists enough to know taxation in some form was necessary for a functioning country - but their issue was that they had no representation in order to vote and argue over said taxation in Parliament like their kin in Britain itself did. See the famous phrase "No taxation without representation". The initial colonial demands weren't "no taxes!" but rather "give us representation, and then we'll discuss taxes". The British government was vehemently opposed to the idea of granting political representation in Westminster to the colonists. They did not view the colonists as being equal to the British subjects actually living in Britain; they were always seen as "others", to some degree. Closely related, yes, but not the same.

The idea that the Founding Fathers were completely anti-taxation, and thus hypocrites for enforcing taxation when they gained independence, is a fallacy. They were only opposed to unjust taxation that was enforced by a political assembly of which they had no representation.

Certainly, by our 21st century standards, the results of the American Revolution leave a lot to be desired in terms of suffrage and rights, but history cannot be viewed from this perspective. It must be viewed relative to how things where at the time, and in the late 18th century, the republic created by the Founding Fathers was remarkably progressive and modern. These men were extremely well read and intelligent, and spent years considering the precise nature of their republic and weighing up the pros and cons. The subsequent chaos and violence of the French Revolution served as proof, in their eyes, that a more populist form of republicanism was a bad idea (though the French similarly did not expand voting rights to women and non-whites). A few decades later, tens of millions of downtrodden Latin American nationalists and patriots would look to George Washington and the Founding Fathers as inspirations for their own wars of independence. They would not have done this had the American Revolution solely been a 'revolt of rich people'. The fundamental ideas espoused by the Founding Fathers were far larger in scope than that, even if the reality was limited by practical political considerations at the time.