I'll answer your questions in reverse order.
I suppose I even have an issue with "the coordinates of particle one" - doesn't that imply that we know the exact position of the particles, when we are trying to find their wave function, which is necessarily spread out.
Just think about a one particle case. We can talk about 'the coordinate of the particle $x$', and the wavefunction $\Psi(x)$ is a complex-valued function of that coordinate. The coordinate is just a label for possible positions -- it does not imply we know that the particle is located at any particular position, since $\Psi(x)$ doesn't need to be a delta function.
I'm having a little trouble with this. I would have thought that wave function (even for a two-particle system) would just have a value for each point in space and in time and so would just be: Ψ(𝑟,𝑡)
This is indeed a subtle point and this is really one of the key points at the heart of quantum mechanics.
When you are first introduced to quantum mechanics, you tend to think of the wavefunction as being "like" the electric field, but instead of a vector at every point in space, you have a complex number. Then you might think the natural generalization to having two particles is to have two fields. For example, by analogy, you might think of having two fields lke electric and magnetic field, and so you have two vectors at every point in space.
In fact this is very much not the correct generalization. A better analogy for quantum mechanics is probability theory. Let's say I want to know the probability of it raining on Monday and Tuesday. You might say, "well there's a 10% chance of rain on Monday and a 10% chance on Tuesday, and I multiply them together." But this is not a good model because these events are correlated. If it rains on Monday, then there is a higher chance that it will rain on Tuesday.
In quantum mechanics, we need to associate one complex number (called a probability amplitude) with every state of the system. For two particles, a state of the system would be "Particle 1 is at location $x_1$ and particle 2 is at position $x_2$." We can write the complex number associated with this state as $\Psi(x_1,x_2)$. You might then think, by analogy with the electric/magnetic field situation, that you could decompose this complex number into a complex number associated with particle 1, and one for particle 2: $\Psi(x_1,x_2)=\Psi_1(x_1)\Psi_2(x_2)$. In general this is incorrect, because we want to allow for the case that particle 1's position is correlated with particle 2's position. Therefore we should really think of the two particle wavefunction on living on a space which is 6 dimensional, 3 dimensions for particle 1 and 3 for particle 2.
Having said that, there are specific situations where it is ok to think of particles 1 and 2 as being uncorrelated, in which case it is fine to decompose $\Psi(x_1,x_2)=\Psi_1(x_1)\Psi_2(x_2)$. This is also true in probability theory -- if we have two dice and want to know the probability that both are a 6, then we can in fact multiply the probability that dice 1 comes up with a 6 and dice 2 comes up with a 6, since these are two independent events. However, it is not true in general.
For what it is worth, I personally feel that this subject is made more confusing because of how it is taught, which is often to introduce the topic with two particle problems where the particles are uncorrelated and the wavefunctions for each particle can be separated. As I've tried to emphasize this is only a special case, and if you aren't careful you can get the wrong idea about how to generalize quantum mechanics from 1 to 2 particles.