I can't think of a catch all term for what you are describing. There are however, some frameworks for analysing sound. I think the range is equally vast, especially if you consider sound design in the context of its original definition - film sound, then the combination of sound and image is very complex indeed.
But I think you are looking for something more quantifiable. First off, I think you should read The Tuning of the World by R. Murray Schafer and Sonic Experience A Guide to Everyday Sounds, whilst not necessarily the most sound design specific books out there, they have a multidisciplinary approach to them which you may find helpful. In The Tuning of the World, Schafer proposes a system for classifying sounds or recordings of sound (which has been developed a great deal since). Duration, Frequency/Mass, Fluctuations/Grain and Dynamics are some terms used here, along with the equally important semantics of sound, if you are trying to analyse how sound might affect someone. Other parts of those books look at the physical effects of sound, such as how vibration from different frequencies affects us physically or how acoustic spaces change our perception of sound.
The book Sound Design by David Sonnenschein also covers many different areas with more of a focus on film sound, so that will also be a good read.
Depending on what you are interested in, Acoustic Ecology deals with analysis of sound, soundscapes and how sound affects us, but less with the design element or how sound will work with image. Sound Design itself is a relatively young field, but there is plenty of information out there. Sonic Interaction Design considers physical interactions and how sound influences these. You also need to bring music theory into the fray, especially with the example you give of a logo, as many of these have musical elements. Whilst I'm not sure drawing direct parallels to vision is necessarily useful, the combinations of colour you mention could easily parallel musical harmony and dissonance.
In short, there is no magic bullet! I'm also tempted to say that you can't approach sound in the same way as something visual as it's fundamentally different due to the time axis, though perhaps I'm just being pedantic here...
I hope some of that is useful to you though. Hopefully others will chip in with their thoughts. Worth also mentioning that there are plenty of answers already here which are relevant to this, so poke around the site and see what you can find.
