Yes, They are cooler, yet larger. The larger the star, the higher on the diagram, and the cooler, the further right.
The nearest stars - assuming all stars up to a certain distance are actually known - are representative of the general star population. The brightest stars are not. They are... brighter than the average stars.
Bottom right
Large stars are distributed all over our galaxy, there is no single location for them.
Depending on the luminosity they will appear to the right of the HR diagram.
upper right
Hot stars are found in the left hand side of the diagram, cool stars the right, bright stars at the top, and lastly the faint stars are located at the bottom.
All stars.
Upper right
They are located to the upper right in the H-R diagram.
The nearest stars - assuming all stars up to a certain distance are actually known - are representative of the general star population. The brightest stars are not. They are... brighter than the average stars.
Bottom right
A red dwarf is located at the bottom right of the H-R Diagram. The stars located in that region of the diagrams are small, dim, and red.
It is found in large amounts in the stars and in the Earth's atmosphere
Upper right
The HR diagram has the star's temperature along the horizontal axis and the absolute magnitude (brightness) along the vertical axis. Each star is represented by a single dot. Higher temperature is usually associated with more brightness so many stars lie on or near a line on the diagram called the Main Sequence. Red giant stars are found on the upper right hand quarter because they are relatively cool but still very bright.
Large stars are distributed all over our galaxy, there is no single location for them.
The HR diagram contains only stars - so everywhere.