All the time I see examples of the normal/Gaussian distribution with continuous random variables. So my question is do all continuous random variables have a Gaussian distribution?
Lots of real life variables have distributions which are better described as other distributions. t-distributions (heavier tails) are common, as are various skewed distributions, for example, many real measurements must be positive, so greater than or equal to zero, but can have a long tail of high values. Quite a lot of real world data is counts, or similar integer data, which is often better described by a Poisson distribution.
In my personal experience, in epidemiology, bio-medicine, and sociology, genuinely 'normal' distributions, that is real data which can best described as a normal distribution, are uncommon, but it does depend on the filed you work in, and exactly what data you are looking at.
Some of the non-normal continuous distributions introduced to new students of statistics include:
The normal/Gaussian distribution is important because of the Central Limit Theorem (CLT), which shows that for very many situations the sum of randomly distributed independent variables will tend to have a normal distribution, regardless of the constituent variables' original distributions. This can be useful for performing certain kinds of commonly-used statistical inference, which probably contributes to the frequency with which one encounters the normal/Gaussian distribution. Student's T distribution mentioned above gives some formalism to the "tend" in the CLT's "...will tend to have a normal distribution...", and is therefore also useful in these commonly used forms of statistical inference.
Not necessarily. The shape of the distribution is contingent on the continuous random variable's PDF - which isn't expressly Gaussian. Some counterexamples include the student-t distribution and the Laplace distribution.