I'm not a modeler. But I'm an aspiring computer scientist, I'll give it a shot without being too technical or going outside the bounds of my understanding.
1 - Why is it the more detailed an object is, the more data and memory it takes to read it?
The more 'detailed' an object, the more triangles/quads or polygons and vertices it is likely to have. Bassically each triangle, etc is a piece of information. The more triangles in an object, the more information that is required to store it and process it on screen.
2 - What affects what more and in what manner?; is it the memory-size of an object that causes the lag or is it the amount of objects in total that causes the lag and what affects the other first?
I'm not sure I understand exactly what you are asking. But, think of it as total amount of information. The more triangles and such you have in a single object, the larger its memory-size will be. Just as well, as you add more objects to a scene, each one of their individual memory-size adds up to the total memory/processing power being used. So, in (oversimplified) theory, a few high-poly models or lots of low-poly models would perform about the same if the total amount of information/triangles is the same. Now, it's not quite that simple once you factor in animations and the overhead of rendering individual objects, but that's going beyond my understanding of 3d graphics so I will stop here.
3 - Is there an alternative to Polygons and Vertices?
Voxels are once that's gotten alot of attention recently. Lookup "Unlimited Detail" and "Atomontage" on YouTube. I won't describe the way they work from memory as I wouldn't want to describe them inaccurately, but voxels are basically ,3d pixels' if that makes any sense. Akin to particles or "Atoms" as the Atomontage guy calls them. As of now, these technologies don't really work very-well for games because of the difficulty in animating voxel objects in real-time. So no, you can't use these. I think it would be amazing if someone developed a hybrid voxel/polygon engine that used voxels for static (non animated) objects like terrain, rocks, buildings, etc and used polygons for animated objects like characters and foliage. I'm not sure what the technical issues would be in implementing something like this though.
4 - If their isn't an alternative, is it possible to create a new form of modelling or said alternative, in which the object can be infinitely more detailed with no memory-size difference and there can be an hundreds of times more of these objects in one scene with no Memory-size difference and lag?
Machines have limited memory, unless you've got some sort of Turing machine on your hands
You can not have 'infinitely' more detailed and numerous iterations of anything represented within a machine without running out of memory at some point. For this reason, research in optimization algorithms is a hot topic within computer science. Sadly, not my area of expertise. But, to answer your question: if you're talking about objects that are placed repetitively throughout a scene, there are methods of optimizing their performance. Describing those methods would be, again, be outside my comfort zone. But, for example, the SpeedTree technology that Oblivion uses for foliage would be making sure that 100 trees doesn't take 100 times the processing power of one tree.
I say I ponder upon this and ask these questions because of my ridiculous vision in scenery within games, especially Morrowind.
That's one of the difficult things about game development. You have to balance your ideal visions against what is possible to do with the current hardware. 3d animated movies can be so detailed because they are allowed to take lots of time (days? weeks?) to render 2 hours of footage on a render farm with many processors. Games have to render in real time on a single machine. But, computers are getting faster every year. 10 years from not we will be able to do much more with games than we can now, just as we are able to do much more now than when Morrowind was released 10 years ago.
Anyway, 'low'-poly modeling is a bit of an art within itself. That is, modeling objects that look like they are more detailed than they really are. I'm not an artist, so I can't give you any tips. But, do research on low-poly modeling for games and practice your skills.
Hope this helps!