What are the pluses and minuses of taking each multiple-mapped tag?

You need to be confident where your sequence comes from before doing statistical analysis, of course you can try to use that method and check for variations in your datasets, just be careful with your statistics.

But with technologies pushing longer high quality reads, the proportion of multi-mapped reads will be irrrelevant soon.

Thus, I am showing the probability that the tagged mapped to that position. Has anybody ever tried this?

Yes, check the literature, this strategy has been used in DNAseq and RNAseq (sorry but I don't have the references in my head right now). Some tools even use this method for initial step before redistributing the values in iterative steps, check Cufflinks related papers.

What Salzberg endorses in this paper (http://www.nature.com/nrg/journal/v13/n1/full/nrg3117.html) is to map a multi-read randomly to one of the mapping positions. This has the following advantages:

• Is faster compared to mapping all positions the read may map to.
• The output mapping file does not need any extra processing besides the usual pipelines.
• Is built-in in some of the aligment software.

Regarding your proposed method I have mostly seen in applied in the context of RNA-seq. Here is a paper on the topic: http://bioinformatics.oxfordjournals.org/content/26/4/493.long