I don't know about the full version; my gut tells me that it has registry and path and OS level crap that prevent you from doing this easily. (You may be able to brute force it with some hackery and supporting programs or batch files to set up these things)
and one other way:
if your external hard drive is BOOTABLE and you can get that to work, then you could do it as the OS changes needed to run VS would be on the disk in its booted OS. Windows used to check your hardware serial numbers and refuse to work if moved off its original installed machine (does it still do this?!), but VS can work on linux I think, and you can certainly boot that off an external drive.
and finally:
if you wanted to do it, you can set up your home computer such that you can just connect into it remotely and use the home PC from the remote PC directly. This is frustrating esp if your networking is slow, but even fast net it feels a little off, but it works.
It it possible to install VS 2017/2019 on a secondary drive, D:\ for example, though the installation(s) do put a lot of files on the C:\ drive.
I installed VS 2017 and 2019 on my D:\ drive. About half of the files were installed on C:\. As long as the external drive is the same drive letter from reboot to reboot I can't see why it should be a problem.
Trying to make a portable install? AFAIK not possible.
VS 2005 is outdated. C++ has changed considerably since 2004.
there is also docker, but I dislike it. It basically makes an image of your entire OS + the program, its incredibly fat. Maybe it has options to make a smarter VM but so far I haven't had much luck getting the bloat down.
I've played around with Docker, and it would be a pig on an external drive, even with USB 3.0. VS2017/2019 is already at times somewhat sluggish and cranky even when installed on internal SATA drive(s). My install is split between a regular platter SATA drive and an SSD. Boot drive is the SSD.