Brownian Motion has a deep link to the heat PDE. Studying the dynamics of a particle moving (as if undergoing Brownian Motion) one can derive the heat equation. Also looking at the generator of the Markov Semi-group of Brownian Motion, we see it is the Laplacian (and by the differential property of the semi-group $d_{t}\mathcal{P}_{t}f=L\mathcal{P}_{t}f$ ) we again see a connection between the two.
Firstly - Applying this link to the real world has it got something to do with heat just being the movement of particles (my physics is terrible haha)?
Secondly - I was wondering if other PDE's each have similar links to some Stochastic Process?
There is the Fokker-Planck equation of an Ito process. Basically every PDE of the form
$$\frac{\partial p}{\partial t}=-\frac{\partial}{\partial x}\left(\mu p\right)+\frac{\partial^2}{\partial x^2}\left(\frac{\sigma^2}{2} p\right)$$
is the evolution of the transition density of related Ito process satisfying the SDE:
$$dX_t=\mu(t,X_t) dt+\sigma(t,X_t) dW_t$$
Letting $\mu\equiv 0$ and $\sigma\equiv 1$ corresponds to Brownian motion and heat equation.
Also related is Feynman Kac.
Also, related to your question about heat and particles - a resounding yes. Check Einstein's or Smoluchowski's arguments. Particles undergo a random walk, randomly bumping into other particles. Because the number of particles is so high we can apply the central limit theorem/Donsker's theorem to get that the macroscopic behavior is a Brownian motion.