Do variant calls change when you call from CRAM? We're considering switching our storage format from BAM to CRAM. We work with human cancer samples, which may have very low prevalence variants (i.e. not diploid frequency).

If we use lossy CRAM to save more space, how much will variants called from those CRAM files change? Which compression strategy has the lowest impact? 

Are there any other impacts on downstream tools that we're not considering?
By default, a CRAM you create with samtools is lossless. It typically halves the input BAM in terms of file size. If you want to compress more, you can let samtools convert most read names to integers. You won't be able to tell optical duplicates from read names, but this is a minor concern. You can also drop useless tags depending on your mapper and the downstream caller in use. For cancer data, I wouldn't reduce the resolution of base quality without comprehensive benchmarks. Unfortunately, base quality takes most of space in CRAM. Discarding the original read names and some tags probably won't save you much space.