Is it USB or Firewire drive? USB is far too slow for sharing of files. And if firewire, firewire 800 would be best.
I had problems with using an external firewire 400 drive on my old computer for sharing of files. Mac OSX didn't seem to be constantly reading the drive so when LW started, it didn't always see the drive. And even if re-adding it after opening LW, had limited hits to my shares from it. That problem might have been partially due to my old G4 bus speed, etc. I have used an external drive for my iTunes library & iTunes itself.
LW does seem to have made some odd changes to the installing of LW. It now requires a parent Application folder directory folder. So if installing onto the external drive, you would need to maintain such a folder there (it seems to have same permissions as the system default application folder.)
Problems with opening a program from an external drive may include the constant need for the system to read (& write) from/to the external drive whilst that program is also accessing the internet. I have a macpro now so I might experiment with one of my externals to see if what you are doing works for me, with my faster system (multiple processors, etc.)
* Keep in mind, some people cannot open the new LW 4.18.x versions on their newer intel macs. I cannot use it on my system, I am presently using LW 4.16.6 (which has some bugs.) If you would like to try an earlier version, send me a
PM (private message).