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(A much shorter explanation than the below wall of text is near the bottom of the comment.)
SMD mesh files only store independant triangles, where the only information each triangle has is the material of the triangle, and each of the triangles' vertices' 3D positions, "normal" directions, UV-mapping, and bone influences (weight-mapping).
Well, of course, SMD mesh files also store a skeleton/armature, with bone positions, rotations, and parenting all kept intact, but the point is that for a flat square, it doesn't store a 4-sided polygon, and it also doesn't store 4 vertices where it says which 3 vertices form one of the two triangles, but instead it stores 2 completely independant triangles.
So when importing SMD mesh files into Blender with "Preserve SMD Polygons & Normals" disabled, the SMD mesh gets imported with basic smooth shading, and then a "remove doubles" thing gets applied automatically, in order to make it so you get 4 vertices that are connected as 2 triangles.
But when importing them with "Preserve SMD Polygons & Normals" enabled, you get 3 vertices that form a triangle, and 3 other vertices that form another independant triangle. Even though 2 of the vertices from the two triangles may share the exact same position in 3D space and such, they will still be 2 independant vertices, so you have 2 different vertices there. This effect just gets "amplified" the more triangle vertices that are there.
(The reason the shading looks fine instead of blocky this way is because it also imports the "normals" of the triangles' vertices when doing this, which for various reasons can't be done when the "remove doubles" thing gets applied when "Preserve SMD Polygons & Normals" is disabled.)
Now, the way VTA flex files work is that they have a "base" position with a vertex in place for each vertex of the model they're intended to affect, and then they have different "vertex animation" frames where those same vertexes/vertices are moved to where they should be in the different flexes.
Both Source's StudioMDL (model compiler) and the Blender Source Tools have VTA flexes work roughly by simply comparing the VTA file's "base" position vertices to the vertices of the mesh the VTA is applied to, and then assigning that mesh vertex to that VTA vertex, having that mesh vertex do the same movements for the various flexes as the same vertex of the VTA file.
As you may remember, having "Preserve SMD Polygons & Normals" enabled when importing an SMD mesh file with the Blender Source Tools makes it so that each triangle is not connected to any other triangle, so there will be many different vertices at the exact same point, while the VTA file only has a single vertex at that point. And even if the mesh has multiple vertices in the same spot, only a single one of those vertices can be considered the "closest" to the VTA's vertex there, meaning that only one of the triangles will have a vertex that moves from flexes there, with the rest of the triangles having their vertices be stuck in place there.
I know you may be wondering why StudioMDL doesn't have this problem, and the short answer is that it simply is able to merge multiple SMD mesh vertices into one where they are in the same spot, while also being able to keep the custom per-triangle shading intact unlike Blender's "remove doubles" thing. The long answer is the same thing, just lengthier.
So if you still don't get why the flexes screw up when imported onto a "Preserve SMD Polygons & Normals"-imported SMD mesh, it's because that setting will make there be multiple vertices in the same spot instead of just one, while the VTA expects there to only be one vertex there.
TL;DR: "Preserve SMD Polygons & Normals" makes each triangle of SMD files be imported separately, not connected to each other, while VTA flex importing basically requires them to be connected to each other.
Again, I don't know why having "Preserve SMD Polygons & Normals" disabled would result in some of the triangles being flipped, but that's kind of the "main" problem you should focus on. The broken VTA flex importing is simply a result of "Preserve SMD Polygons & Normals" being enabled.
Try to force Double-Sided rendering, find all flipped normals and flip them manually, or use an older version of Blender to import the models then work on them in the newer version.
EDIT: Then go to the Object Data tab and either uncheck "auto smooth" or turn it up to 180. Unless you want it to do it, of course.