Removing green Xs and red lines in QGIS?

You are in the middle of editing or creating a new line feature. If you toggle the layer visibility you still see the half-created feature, and if you toggle editing off in that layer you'll still see it!

If you re-activate the layer visibility of whichever layer it is, and re-enable editing, you might find you need to finish creating the element before saving the layer. Delete the feature if you don't need it.

The green crosses are self-intersections that show this is an invalid geometry - you should also have seen a warning like "Add feature: segments 0 and 2 of line 0 intersect at -2.35695756492, 0.44230006192 Geometry has 1 errors. Validation finished"