Generally, there's no inherent problem, in QTL mapping, with using very dense markers, as the increased density is accompanied by increased association among markers. LOD curves will be a bit more wiggly, but not very much so. As you increase the density of markers from 20 cM to 10 cM to 5 cM, you do increase your power a bit, but beyond that, not so much.
But it is important, in the R/qtl software, to provide cM locations for markers rather than basepairs locations. On your map, the range of marker positions should be on the order of 100s not millions. Having the markers in basepairs may lead to noisy results, but I wouldn't think so, unless maybe your marker genotypes have a high rate of errors.
The skewed distribution of the phenotype *does* make it harder to identify QTL, but I'd expect lower power and low LOD scores, rather than particularly noisy LOD curves with lots of peaks. It will be very hard to map a binary trait in an intercross with 7 1's and 116 0's.
karl