"Reading between all the lines I believe that all the OSes will be on the same VM, hence in the Virtual Machines folder in subfolders. (e.g. vserver, vwin7, vwin8) Upon launching the VM, the library will list all the installed OSes and offer me the opportunity to open one (or more of them to run simultaneously). Is that correct?"
VMware creates a master folder in documents per default. It then defines a subfolder for each system that you install. You can move/copy the master folder or subfolders to any place you like. But you can also define your own folder for a system during the OS installation and put that wherever you want.
"If I copy the folder (leaving a copy on the internal drive), I can create a desktop shortcut to the folder or vmx file on the external drive to launch the VMs. Is that correct?"
Nah, you have to "Open a virtual machine" in the VMware player control window (open the .vmx). Then you start your virtual system from there. Click on it (top left) and then "Play virtual machine" (bottom right). Shortcuts on the desktop won't work. But you can make a VMware Player shortcut on the desktop. That will bring up the VMware control window.
"For example, if the host has only 50GB free space, there may be issues. I foresee the possibility of failure to launch and run the VMs, or worse yet, overwriting files on the host system. Do I have the correct read on this?"
That's not going to happen. And if you run the OS from an external drive, there is no space taken on the host partition.
"Am I correct in thinking that the virtual network would experience the fewest problems if I use a private 10.x.x.x addressing scheme for communication within the virtual network?"
I don't have the faintest idea. I never tried that.
Here again, if you want to run the virtual systems from an external drive, don't even bother with a HDD. You will not like it. SSD is the only way to go. If you don't want to buy an SSD, run it from the internall HDD.
If you are quick, you can get a cheap 256GB SSD today 3/27 at Newegg. $159.99 With Promo Code: EMCYTZT31567.
PS. Next time you have multiple questions, please number the questions. On this retarded forum you cannot make proper quotes, so the answers get very messy.