If climate change gets out of control, will the US try to annex Canada?

[Ban my account ffs!]

No we shouldn't and we wont annex Canada.

The great majority of climate science papers that predict the most gloomy outcomes by the year 2100 rely on the RCP 8.5 scenario as laid out by the IPCC in its most recent Climate Assessment Report in 2014.  RCP stands for Representative Concentration Pathway and the number refers to the amount of the increase in energy at the earth's surface as a result of human activities by 2100 in terms of watts per meter squared.  So this scenario, being one of four drawn up by the IPCC, represents an increase of 8.5 watts per meter squared of energy at the earth's surface, resulting in a warming of ~3.7*C in the average global temperature.

There are 3 other scenarios, RCP 2.6, RCP 4.5, and RCP 6.0.

Each scenario was proposed in terms of the amount of forcing that human activity would cause on our climate by the year 2100 (meaning they simply picked the 2.6, 4.5, 6, and 8.5 numbers out of a hat based on the likelihood of the climate reaching these thresholds by 2100).  The RCP 8.5 scenario, which most heavily publicized studies by the media focus on, is actually very unlikely to occur.  The timeline of policy actions by the global community assumed to reach the RCP 8.5 scenario are not realistic, and break away from many long-standing trends in human development (including some that have been improving for over a century).

RCP 8.5 assumes high population growth to 12.3 billion, almost no further action to reduce emissions, slow economic growth (25% less than the RCP 2.6 high mitigation scenario), a dramatic slowdown in technological innovation, a reduction in energy efficiency resulting in huge increases in electricity production, and a significant increase in the proportion of energy that we derive from coal.  (rising from ~45-60% over the next 80 years).

It is basically the "f**k it, we're gonna burn the sh**t out of that coal" and stop innovating technologically!  scenario.

When you simply extrapolate the current trajectories for these factors from now, you end up with a scenario by 2100 somewhere between the RCP 4.5 and RCP 6.0 scenario, which predict a warming of 1.8-2.2*C.  In a high mitigation scenario like those being proposed by individual states and governments around the world, the RCP 2.6 scenario becomes likely and we would be able to keep global warming at a total of 1.5-2*C from pre industrial times.

[Ban my account ffs!]

As for what would likely happen to our ecological biomes in a world that is up to a few degrees celsius warmer:

None of these but the RCP 8.5 scenario pushes us beyond the warmest era of the previous interglacial, which was ~3*C warmer.  Global sea levels were higher during this time (as shown by the stranded fossil coral reefs all over the world) and Scandinavia was an island.

This warmer climate did not feature the vast desertification or uninhabitable tropics that some of the scare stories claim will happen.  In fact, the tropical forests of Africa were much larger.  The Sahel was a seasonal tropical forest and the Sahara was mostly savanna and warm grassland.

Tropical forests weren't as plentiful in South America because the Andes Mountains would act in such a way that the increased trade winds due to the warmer climate near the equator would promote heavier, more frequent precipitation near the mountains but suppress thunderstorm formation in northeast Brazil.

In North America the forest/tree line was further west near Lubbock Texas instead of Dallas and temperate forests reached much further into Canada and the deep south had more of a savanna climate likely due to less fall/winter/spring precipitation and more summer precipitation but also more evaporation, promoting warm grasslands and woodlands.

Western Alaska would've had much more forest cover but the interior reaching into the Yukon featured more of a "northern plains" climate because while warmer weather promotes more precipitation, the mountains would still block much of the precipitation from Gulf of Alaska storms.

In the west, coastal California would be drier and warmer... but the interior west would have much more monsoonal precipitation in the summer, leading to warmer scattered woodlands, grasslands, and desert vegetation areas.

Basically:  Agriculture would have to adapt but we wouldn't be sh**t out of luck.