Great questions! I've lived and worked in Michigan my whole life, and I've done a good deal of professional and academic research into Michigan history, and I studied the War of 1812 for my master's thesis, so this is right up my alley!

First, let's separate your two main questions: what was daily life like for citizens of the US under British occupation in Mackinac and Detroit, and what was the political situation of the indigenous groups that lived in what became the Michigan territory following the war? As we'll see, both of these are quite different.

Occupied Michigan

Michigan was one of the earliest targets of British action in the War of 1812. On July 17th, one month less one day after the United States declared war on Great Britain, a ragtag group of scruffy British regulars - a single under-strength company of the 10th Royal Veteran Battalion - with hundreds of Native volunteers that included locals and Sioux from what's now Minnesota, stole a march on Mackinac Island, surprised the American garrison, and forced a surrender. A month later, on August 16th, 1812, General William Hull surrendered the garrison at Fort Detroit in the face of a similar force of British regulars, Canadian volunteers, and Native allies. Only a day earlier, unbeknownst to the garrison at Detroit, a column of evacuating soldiers and civilians was attacked and destroyed by Potowatomie allied to the British outside Fort Dearborn - modern day Chicago.

Given that Michigan was meant to be an important part of the aggressive American military strategy, this was an enormous setback to US hopes in the theater. But the US military was not, necessarily, totally in step with the needs and wants of the civilians in the region. Michigan was a bit of an odd place, even for an early American territory. Fur trade posts around the Great Lakes had a longer European presence than many other places in territory the United States claimed, with occupation of Detroit dating as far back as the 17th century, and forts at Mackinac and around the Grand River on the west side of the peninsula, and the eastern shore of modern-day Illinois occupied primarily by semi-sedentary indigenous traders (primarily Huron, Odawa, Ojibwe, and Potawatomie) and French fur traders. The region remained predominantly French-settled even after the French and Indian War and the War for Independence. In 1805, William Hull was appointed by Thomas Jefferson as the governor of the Michigan territory, and arrived just after a terrible fire destroyed a great deal of the town.

Even with Hull and the American bureaucratic infrastructure installed, a major proportion of the white settlers in the Detroit and Mackinac regions were of French or Metis extraction. Families of French and Native ancestry were extremely common, as inter-marriage was a terrifically effective means of connecting French traders to local indigenous kinship and trade networks, and had been a part of French trade policy, officially or otherwise, dating from the first permanent French settlements in North America. Ohio militiamen who marched to Detroit before it fell in the first campaigns of the war even remarked that it was sometimes difficult to tell some men of the Michigan militia apart from their Canadian and even Native American counterparts. Quite a few of the region's settled American Indians lived lives extremely similar to the white settlers; Little Turtle, a Miami war chief and prominent leader of the Wabash Confederacy that utterly destroyed two American armies in the 1790s, lived in a Miami village near Brownstown, Michigan, that included brick and clapboard houses, and contained livestock pens for steers and dairy cows. Little Turtle was by no means the only prominent American Indian to live in this style, and given the very tight diplomatic, familial, and trade ties to the long-established Detroiters, many of those considered "white" in a legal sense were likely to have Native ancestry or Native wives and children.

In addition, the question of slavery was a complicated on in the Detroit region. Quite a few British traders and officials kept enslaved servants, but when they moved when the Americans took the city in the 1790s, they took their enslaved people with them. Some Americans - those who settled in the 1790s or came with Hull ten or fifteen years later - brought enslaved workers with them, and some French and Indigenous traders kept enslaved laborers, but there were few in comparison to other regions, owing partially to the kinds of New York quakers and deists who took positions in Michigan in those early years, with whom objections to slavery ran quite high. Hull himself was opposed to it, and went as far as to allow the formation of a company of freedmen militia under a freedman named Peter Denison. Denison aggressively recruited men for his company and allegedly led canoe-borne raids across the Detroit river to free those enslaved by British plantation owners and traders. We unfortunately know very little about Denison's actions in the early campaigns, although his cohabitation with Elijah Brush, the leader of a supply column that was the cause of two skirmishes between Ohio and Detroit in July, 1812 might suggest that Denison and his company were involved in the supply run.

Given this rather complicated social picture of the region, it might not come as much of a surprise that the British were greeted without much consternation. British leaders at Mackinac and Detroit were eager to exploit the civilians' connections to the local indigenous groups to help rally more and more native warriors to their cause. The strike at Mackinac had only been possible because a local fur trader, Robert Dickson, had been actively championing the British cause for months. Others on Mackinac Island, like Michael Dousman, were possible vectors to even more warriors.

The thing is, apart from a mostly unremarkable call for American citizens to swear allegiance to the king to continue operating, very little would have changed after the seizure of Mackinac and Detroit. Mackinac was of course reliant on supply by boat, but British control of the upper lakes for most of the war guaranteed that their supply lines from the Nottowasaga river were mostly cleared. An American blockade of the river by two brigs, the Tigress and Scorpion, was cleared by a daring cutting out expedition and kept the garrison and civilians from being starved out.

One of the more prominent stories about the years under occupation - there is very little about it, suggesting that it was for the most part business as usual - was the tale of Ambrose Davenport, the so-called "Yankee Rebel," who refused to swear an oath to the king and was tossed into prison for the duration of the war. He allegedly declared, on being asked to swear to the king, "I was born an American, and at all hazards I plan to live and die as one!" Others were a little less sanguine in their patriotism, and life on the island was likely a somewhat regular rotation of gift-giving (assembling yearly treaty annuity payments and extra gifts of food, cloth, weapons, and tools to bring more natives to the region), and trade (which continued on a fairly regular cycle and even by 1812, the bulk of the actual trapping was happening much farther west than Mackinac Island, and so the war didn't interrupt it much). Mackinac was only interrupted in 1814 when the Americans attempted to recapture the island, and failed, which led to the debacle (or triumph, depending on how you arrange your reds, whites, and blues) of the Tigress and Scorpion.

Detroit saw its most dramatic action very early in the war, and a good deal of the prominent men, including Hull and his family, were captured and marched to Quebec, to be released on parole soon after. This depopulated the immediate environs of most of the most committed Americans, and the fort continued unbothered for most of the rest of the war, with the same stricture applied to its civilian population: swear allegiance to King George, and keep working as if nothing changed. The British also, of course, organized those volunteers and others from around the region into their own version of the territorial militia, the Michigan Fencibles, but they saw very little action. The Battles of Frenchtown and the River Raisin involved mostly men from Indiana and Ohio, and occurred far enough from Detroit that it was likely most of the civilians in Detroit were largely unaffected by it.

So to wrap this part up: life in Detroit and Mackinac, for the most part, continued in much the same manner as it had before the war, but lacking the American political apparatus that was itself a fairly new structure. Most established traders in the region may have casually switched allegiance from Britain to the US just a few years before, and so switching back was really no bother.

There's a lot more I wish I knew: what happened to the black militia? Denison, unfortunately, died in Quebec when he was still a prisoner, but what about those allegedly emancipated enslaved workers from the Canadian farms? Who were the men who served as Michigan fencibles? How did the Fort Dearborn massacre affect Detroit trade networks? But for the most part, we know that life changed only as much as the war asserted itself in the region, which was, after the drama of the opening months, not much.

While you wait for the second half, you may like to read a couple of my older answers, about the war experience in and around the Great Lakes and what it was like for POWs during the war

Part 2 and sources, below!