Is there any alternative characterization of sparsity of a signal in compressed sensing
Indeed, there are ways in which sparsity, or information content, may be estimated
at the acquisition device. The details, practicality, and actual usefulness of doing so
is debatable and heavily dependent upon the context in which it is applied. In the case
of imaging, one could determine areas of an image which are more or less compressibile 
in a predetermined basis. For example, see "Saliency-Based Compressive Sampling for Image Signals" by Yu et al. In this case, the additional complexity requirements placed on the acquisition device provide marginal gains.

With regards to your questions about making determinations as to the usefulness of Compressed Sensing on a given signal at the time of acquisition: If the signal in question adheres to 
any kind of model known a priori, Compressed Sensing is possible. Accurate recovery is 
simply dependent upon ratio between the number of measurements taken and the degree to which
the sampled signal adheres to your model. If it is a bad model, you won't get past the phase
transition. If it is a good model, then you will be able to calculate an accurate
reconstruction of the original signal. Additionally, Compressed Sensing measurements are, 
in general, future proofed. If you have a given number of measurements for a signal which 
are insufficient in number of accurately recover the original signal using the model you
have today, then it is still possible to devise a better model tomorrow for which these 
measurements are sufficient for accurate recovery.

Additional Note (edit):
The acquisition approach mentioned in your question sounded quite near to adaptive Compressed Sensing, so I thought the following might be of interest for the readers of this question. Recent results by Arias-Castro, Candes, and Davenport have shown that adaptive measurement strategies cannot, in theory, offer any significant gains over non-adaptive (i.e. blind) Compressed Sensing. I refer readers to their work, "On the Fundamental Limits of Adaptive Sensing" which should be appearing in the ITIT soon.