The first thing I'd try is the Densify tool to add coordinates to your original dataset (before projecting to Sinusoidal). I think what might be happening is that you have long line segments without intermediate shapepoints. When projecting, the line endpoints are projected, but the space in between isn't. Anyway, try that first. Maybe start with 10km distance and see what happens. If it looks like it will solve the issue, you can decrease the distance for finer resolution. If this isn't the issue, I'll investigate further.
Which version of ArcMap?
Can you share one of your KML files for me to try?
I don't see your image 2 as problematic:
it looks like a reasonable representation of the line in image 1
and in Google Earth
You likely want to split your track wherever it crosses the 180° meridian,
so Arc will not wrap the line back across the whole earth to continue the single feature.
We know that the world is continuous across that meridian,
and thus the feature should break there when projected,
but the feature coordinate stream tells Arc that the line goes from hither to yon,
and the only way Arc knows to get from hither to yon
is right across the face of the flattened earth.
Yes, Arc is making some unfounded assumptions about how the line should run,
but your 'edited' version is only a reflection of your own unfounded assumptions about how the line should run.
Split the line, and see if you two can come to some sort of agreement.