If a star is brighter, does that mean that it is hotter?
A brighter star is not necessarily hotter.
Bright main sequence stars tend to be larger and hotter.
The notable exception to this are red giants. These are stars approaching the end of their life which have expanded to a great size. A consequence of this is that the surface of a red giant is relatively cool.
Another consequence of being very large is that red giants emit a lot of light and are quite bright.
Some of the brightest stars, such as Sirius and Rigel have surface temperatures of 10,000K or more. Betelgeuse on the other hand is a bright red giant with a surface temperature of less than 4,000K.