Does Thompson's atom model have a solid positive material, or a positive "cloud"? I understand that the explanation of why Rutherford's experiments disproved Thompson's model is that they expected all the alpha particles to go through the space between atoms with minimum deflection.
Does that mean that Thompson's model was a positive cloud with electrons spread uniformly in it? Why didn't he assume it for example to be a solid positive material (even though "solid material" doesn't make sense at the atomic level)? This would have maybe explained Rutherford's experiments as showing collisions with that solid part?
 A: So there were others at the time especially in the German tradition who would have liked that immensely, but that was not J.J. Thomson's tradition. (Side note, there is no p in his last name.) 
In the 1800s and continuing into the early 1900s the theoretical belief in atomism was somewhat of a contentious one. Chemists especially tended to really like it -- this traces back to the work of Dalton in the early 1800s. The basic argument was that if you weighed chemical reactants before and after a reaction, you would not just see that the mass before and after was the same: but you would see that when you separated out the resulting substances those masses generally had a nice whole-number relationship to the original masses. Maxwell had worked out the mean free path of air as around 60nm, providing an upper bound on the size of the atom, since it was believed correctly that gases must involve molecules traveling many times their size before interacting with each other. Afterwards Loschmidt had worked out based on the condensation of gases that these molecules might be about 1nm large; we now know that he was off by a factor of 10 or so. 
In fact the wavelengths of visible light were also known and Maxwell knew in an 1873 talk called Molecules that his mean free path was about a tenth the size of the wavelength of light, so that atoms must necessarily be invisible to any microscope.
There was a not-so-well-documented school of German scientists who were comparatively skeptical of these ideas of invisible particles making up matter. They may have believed that continuous matter could be understood in terms of whole-number solutions to differential equations -- so perhaps it was not so surprising that masses came in whole-number ratios. Similarly that someone had identified a length scale might not mean that any particles traversed that length scale -- or maybe it was a length scale for, say, a vortex in a continuous medium. They would certainly have spread the positive charge over some entire substance.
But Thomson was part of that Scottish/English/Irish school of scientists that firmly believed in atoms and indeed Thomson himself had stuck his neck in by discovering the electron, and so he was beginning to unpack the inner structure of atoms. In doing so he saw almost no reason to abandon his predecessors and suggest that maybe atoms did not exist; rather he just saw his experiments with cathode rays as somehow identifying a constituent of said atoms. In fact his biggest failure in the plum pudding model -- one he was called out relatively quickly on by rising star Arrhenius, who had been discovering the atomic theory of acids and bases -- was in assuming that the electrons he had discovered were all that atoms were -- in other words that the "pudding" of the plum pudding was massless and was just somehow a property of how these electrons were being connected together into an atom. 
But note that neither of them are really involved on a dispute about the existence of atoms in that paper. They are both in agreement that atoms do a great job about describing several macroscopic phenomena. Arrhenius is just pointing out "hey atoms have got to be something more than just a bunch of these electrons in a bag together because that doesn't explain why they weigh thousands of times more than electrons, or why their weights are not strictly increasing as we travel the periodic table."
