This varied largely by place, time, and the conditions of each specific loyalist. To answer your second question first, there were not white loyalist bands in the backcountry that continued to fight the war after the British Army pulled out. It is worth noting that this is exactly what happened with the Iroquois and other native peoples who supported the British.

White loyalists had the options to either integrate back into American society or evacuate to Canada, where they British set aside land for loyalist refugees. This choice was not necessarily permanent. While all of the states kept laws on the books barring loyalists from returning to the country after the war, those laws were enforced with varying degrees of rigor and were eventually repealed. Family and personal connections were key to this process, and wealth, as ever, opened a lot of door. If a person could prove their worth to the United States, and had enough solid, upstanding citizens vouching for them, then rentry into American society was possible.

Early 19th century New York City was dotted with returned loyalists and their descendants. Mayor Cadwallader D. Colden (in office 1818-1821) was descended from a family that sent several officers to loyalist regiments, and spent the war in London. Peter Van Schaak, a lawyer and one-time roomate of John Jay, was welcome to tutor law students in New York, but had to keep a low profile in his hometown of Kinderhook. A smattering of DePeysters, a whose sons had all fought for the British, worked for the John Jacob Astor and other merchants, using their contacts in Canada and England to sell and smuggle American goods around the British empire. So long as they kept their politics to themselves and did not bring up the past, loyalists could be welcomed.

On the Niagara border, "late loyalists" - Americans discontent with the pace of growth in the US and seeing greener grass on the other side - moved into Canada, while loyalists wishing to return to America filtered into the new towns being constructed in Western New York and Pennsylvania. Again, so long as no one discussed the past, loyalists could live side-by-side with veterans of the Continental Army.

During the war of 1812, Canadian provincial units were often led by veterans of the American Revolution or their descendants, and were happy to recall their successes in the last war. This might explain the generally good performance of Canadian troops during that war.

Sources:

Alan Taylor, The Civil War of 1812

Maya Jassanoff, Liberty's Exiles

Valerie H. McKiton, From Loyalists to Loyal Citizens: The DePeyster Family of New York