Do planetary surface temperatures change in unison in a solar system? Are there any known correlations between the changes in planetary surface temperatures in a solar system?

If so, do the farthest planets have smaller albeit correlated changes?
The simple answer to your question is yes. Taking a simplified equation from Carroll &amp; Ostlie, An Introduction to Modern Astrophysics Second Edition, the temperature of a planet can be estimated as:
$$
T_{p} = T_{\odot}(1-a)^{\frac{1}{4}}\sqrt{\frac{R_\odot}{2D}}
$$
Where $T_p$ is the predicted temperature of a planet in a circular orbit of radius $D$ with an albedo of $a$ around a star with a temperature of $T_\odot$ and a radius of $R_\odot$. If the energy output of the star were to increase, raising $T_\odot$, then there would be a corresponding increase in the temperature of all planets orbiting said star.

In practice there are factors which can make this correlation difficult to measure. The albedo  of a planet during the course of a day can vary greatly and the distance of a planet from the host star changes throughout the year. This equation also assumes that the planet is a perfect black body which most are not which can also change a planets temperature and obscure any changes caused by the host star.