The DRC's job is to alert you to features which may be suspicious, but whether or not they will impact your design depends on the context of the error it has located. The errors you're running into are pretty common, see below for description and recommendations.
1) Angle error - This error can often point out unintended routing, since most intended angles will be 45 or 90 degrees. The little lines you are seeing usually happen when you move the part around (as you've mentioned), or when clicking multiple times when trying to route to a pad. While these aren't ideal from the perspective of keeping a clean layout file, they won't typically impact your design, since the V-One software actively attempts to filter out these artifacts.
2) Clearance error - This error pops up when two features are closer than the minimum limit of in the DRC - for pads, this is set to 0.6mm, and is intended to catch IC pads that have a pin pitch below 0.65mm. However, in this case, you should be in good shape - looking at the LSM303DLHC datasheet shows that the pin-to-pin pitch is 0.8mm, with a pad width of 0.5mm. The DRC is throwing an error because the gap between the pads is right on the edge of the 0.6mm cutoff in the DRC settings, although your design should be within spec for printing with the V-One. In this case, the DRC erred on the side of caution.
3) No vector font - this error comes up when the DRC detects text that is not written in EAGLE's vector font. To fix this, right click on the text, select 'Properties', and change font to 'vector' (default is 'proportional').
Hope that helps!