No such thing as a "right" temperature for the world.
Global temperatures have been fluctuating ever since year dot.
By "right", I'll assume the meaning to be "optimum".
For humans, the optimum air/environmental temperature for healthy life and longevity is between 14C and 22C, aprox. even though the body temperature is on average 35C.
The reason for this is that metabolic rates change as temperature changes, and prolonged heating over 35C causes hyperthermia, and increased metabolism resulting in dehydration.
Temperatures below 14C will result in hypothermia over time.
For other animals and plants the variables can be greater.
Anything higher than 22C becomes uncomfortable for humans as human max. optimum air/environmental temperature is no more than 24C, where one is starting to sweat (dehydrate) profusely.
This all depends on one's ethnic physical constitution, so folk originating from the tropics tolerate higher temperatures than those living in more temperate zones.
If world temperatures rose by another say 5 degrees, humans and all animals would largely migrate to more optimal areas, but plants would either die out or adapt.
It's the plant base dying out that will cause the biggest knock-on effect in global warming, and a higher global temperature would also see an increased CO2 level in the atmosphere and a decreased CO2 level in the seas along with increased salinity at greatest depths contrasting to a reduced salinity at surface levels due to ice-melt.
Also the resulting coriolis effect and El Nino/La Nina cycles would also be altered markedly manifesting in greater changes and extremes of weather patterns.
Increased CO2 levels will see more carbon fixed in plant structure as sugars and less water used by plants, resulting in a greater humidity level in the soil, so effecting water run-off and disrupting other plant and animal life cycles.
Life performance of animals feeding off this increased sugar, low protein, high yield greenery are altered to their detriment, metabolic rates change and reproduction is affected.
Higher salinity levels which have been noted over the last few years (whereas they have been pretty stable for millions of years) occur at depth, while fresher, warmer water at the surface increases. This affects oceanic circulation patterns causing the coriolis effect to also change, resulting in changes in weather patterns.
This higher salinity level and lower CO2 level in oceans causes less calcium carbonate to form and be taklen up by sea animals, resulting in more aragonite rather than calcite shell formation in moluscs etc, so altering the life forms as they adapt to this change. Those that can not adapt become extinct, as occurred during the last extinction phase at the end of the Cretaceous.
Mass extinction events, are linked to unusually rapid global climate changes, warming and cooling cycles and changes in CO2 levels within a general overall cooling trend towards a new ice age, as is occurring now.
As temperatures start to lower again, CO2 is once more increasingly absorbed by the oceans reversing the process to form a calcite sea once more, but a world occupied with a whole diferent set of life forms which again slowly adapt from aragonite to calcite shelled sea creatures and the atmosphere slowly becomes more humid.

Graph from;
Cretaceous
^ Extinction phases over the last 542 million years.
The last one was around 66 million years ago, the cretaceous-paleogene (KPg) extinction phase.
This is my understanding of the process of global climate changes.