Proper way to handle diffuse + refraction + reflection rays in path tracing? Let's say we are path tracing and that there is an object which has some amount of diffuse reflection, and some amount of specular reflection.

What is the best or correct way to handle that in the path tracer?

Looking at smallpt (http://www.kevinbeason.com/smallpt/), it doesn't allow for objects to have both, but instead forces materials to be either diffuse or specular.

I've seen other path tracing code which rolls a random number to choose which way to go, using the luminance of the diffuse and specular reflectance as weightings in the random roll.

I've also seen code which follows both paths and diverges when this happens and sums the results, where each recursive ray was multiplied by it's corresponding reflectance.

I've seen refraction dealt with similarly to all the above.

Is there a physically correct way to handle these situations?
Let us take one step back.
When you do path tracing, you are doing a Monte Carlo integration.
What does this mean? You try to solve $\int f(x)\mathrm{d}x$ by sampling.
The Monte Carlo integration says for a sufficiently big $n$: $\frac{1}{n}\sum_{i=0}^n f(x_i) \rightarrow \int f(x)p(x) \mathrm{d}x$

If we instead sum $\frac{f(x_i)}{p(x_i)}$ we get $\frac{1}{n}\sum_{i=0}^n \frac{f(x_i)}{p(x_i)} \rightarrow \int f(x) \mathrm{d}x$.
Now your $f(x)$ is the rendering equation and you got yourself a way to approximate the rendering equation.

Now that we are on the same page about the basics, let's consider what you do.
You generate a path of light, that will become one of the samples $\frac{f(x_i)}{p(x_i)}$. In order to do this you model the interaction of the light with the surface at each step along the path. And you always update your current path probability $p(x_i)$.

What is now the correct way to handle a material that has multiple different ways that a ray could travel? There is no one correct way. The only important part is, that your values $n$, $f(x_i)$ and $p(x_i)$ stay correct.

When you diverge the path and spawn two rays, you have to increase the number $n$ of total samples accordingly because now you have done two samples. Then handle both pathes as if they were separate from the beginning.

If you want to only continue one path, you can do it with the russian roulette way.
Define three intervals (one common way is to chose the intervals to be a partition of the numbers between 0 and 1). The relative sizes of these intervals have to be proportional to the material. The tricky part is how do you design this. In my class we defined a number reflectivity $\rho$.
We draw a random number between 0 and 1, if it is lower than $\rho$ the path ends. Else we reflect. The BRDF will take care to weight the specular vs diffuse part if you do enough samples.

Of course you can improve this via importance sampling, and that is probably where you have problems. For a very specular surface design a random distribution that just draws more directions from the specular reflection direction. Rest will be diffuse. I don't have enough experience with actually bulding path tracers to help you how to design one, but your sources will probably have something here.

Important here is again: Design it so that you know how probable a specific direction is, and then keep your $p(x_i)$ up to it, so you can in the end divide by this and get a correct Monte Carlo estimate.

TL;DR There is no correct way. It is ONLY important to keep the probabilities of samples in mind.

I hope I haven't missed the point of your question entirely.

Addition:
Monte Carlo samling, as long as you correct weighting by probability, will always with infinite samples converge against the correct value. This means even if you use a complete stupid way to generate samples, if you do it long enough, that will work. Maybe not in your lifetime or that of your great grandchildren, but it will.