Analysing the residuals themselves As far as I know, it is possible to fit a linear regression model and then fit a second model to predict the residuals from the first model by using some other variables. By this you can understand their influence on the relation modelled with the higher level model. So the purpose is not to check the model fit, but to get a deeper understanding of the data. Unfortunately, I was not able to find any literature on this.
Is there a name for this kind of analysis? How and why is it done? What are the pros and cons? Could you provide any literature on this?
 A: This has been referred to as residual index, although not consistently. I guess the type of analyses you subject it to would depend on your question of interest (as most result in some level of 'deeper understanding'), and so would pros and cons. Garcia-Berthou discusses cons of one example of such application as "an ad hoc sequential procedure with no statistical justification" here http://onlinelibrary.wiley.com/doi/10.1046/j.1365-2656.2001.00524.x/full 
In other words, if you suspect other factors are affecting the response, why not start with a model that would account for these multiple factors and their interactions. Yet, in other cases it is possible to justify, and there are valid examples of its use in spatial analyses.
A: As a partial answer to your question,  you can build a residual model (also known as a variance model) to
model the residuals for the original model as a function of the predicted response (for example) as follows (in R notation):
residual.model <- 
    lm(abs(residuals(original.model)) ~ predict(original.model), ...)

You can then examine the residual model to get a deeper understanding
of the data.  From the residual model you can estimate prediction
intervals, for example.
The residuals are usually much more noisy than the original data
used to build the model, and so there will be more uncertainty about the
residual model than the original model.
The above residual model uses lm and thus assumes (at most) a linear relationship between
the absolute residuals and the predicted response, which is often a good enough
approximation to the (unknown) underlying reality---or maybe I should say, a linear residual model is often about the
most complicated residual model you would want to use given the noise in the
residuals.
My 
earth
R package builds variance models essentially using the above idea
(but the ideas are fairly universal and apply not just to earth/MARS
models).   Some of the background theory can be found in the package vignette 
Variance models in earth.
When reading the vignette, mentally substitute the name of your model
instead of "earth" e.g. substitute "lm" for "earth".
Additional references can be found in the above vignette.  Especially helpful is Carroll and Ruppert 
Transformation and Weighting in Regression.
