Most of export formats from 3dsMax or Maya produce split vertices, so every triangle gets its own 3 unique vertices. However, some of these vertices are absolutely identical (position, normal vector, uv mapping, color, rigging, etc.). If you toggle an import option "Optimize vertices" you are likely to get a model with nearly the same mesh as you had before. Some limitations apply like edges on shape where uv-mapping differs, but it's oK.
Also, I can see a random lines in viewports, it looks like your hardware bugs the rendering. this usually happens on laptop devices when ZModeler uses embedded video card (e.g. on-chicp Intel Graphics). If you have another video card in your laptop, you should configure this video card to be a preferred for ZModeler application. I had to do the same on my Acer laptop, so ZModeler can finally "see" an availability of GeForce card. I do this in nVidia control panel tool.
It sounds like the mesh's vertex welding got lost during import. Try using the "Merge Vertices" or "Weld" function in ZModeler to fix the separated faces. Also, check the import settings for any options related to normals or vertex welding.