Lets approach this part by part.
Will the entire Ubuntu software fit onto a single CD?
This depends upon how you define "the entire Ubuntu software". Generally speaking, the software archive is over half a terabyte in size as of April 2013. While there are tools to mirror portions of the archive, it is currently considered impractical to have it available in whole on a single disc.
As to the installation image for Ubuntu, it does not currently fit on a single CD. The last release that can be located for example via the mirror in Portland shows the 12.04.3 release brushing dangerously close to the capacity limit for a CD. Generally a DVD or USB key will be needed for installation use currently.
If I download Ubuntu to a CD, for use on another computer, will it automatically install/partially install on the computer I use for the download?
No. Downloading the disc image does not trigger installation on the downloading computer.
If alternative measures must be taken to prevent installation on the 'host computer', what steps must be taken, since I do have access, but do not have permission to download to the 'host computer'?
As you've noted elsewhere in a comment that you do not have permission to save content to the computer, you may be best served by any of the following options: purchasing an installation disc from Canonical, locating a local Ubuntu social community that may be able to connect you with an installation disc, or consult third-party vendors such as on-disk.com and OSDisc.com