Try ticking the QuickTime scene preferred setting in the color profile options in the comp. Something like that. Red is always a hard color to export as well, at least in my experience.
This is a common issue when going from AI to AE. I’d assume that the white lines are where the swatch edges are. Your best bet is to extend the pattern to the width you’re looking for. You will also see this if you have two shapes stacked in AI that are identical sizes. When bringing that into AE the bottom color will bleed through on the edges. I usually chalk this up to going from vector to pixel format. Patterns can also be made easily in AE using the repeater feature in a shape layer.
There is a script in AE called “Create Nulls From Paths”. It should be under “Window”. Click on your path and then select Points Follow Nulls and it will create Nulls wherever you have anchor points on your path. Then you can either parent the circle to the null or vise versa. I believe there is another free script out there that enhances this one so you can also animate the handles of an anchor point. It is called “Create Nulls From Paths with Tangents”.
Try setting your color working space to sRGB 2.1 and tick the box on for “Compensate for Scene-referred Profiles”. You will find that in the Color tab in Project Settings. There is a slight gamma shift when exporting to .mov files. You can read about that by googling it. No real way to fully compensate for it except with those setting I mentioned. It should at least match better what you have in AE to VLC. This is if your color is off slightly throughout the video. If its just the beginning section it could be something in your project.
If you are using Window Media Player this is a common occurrence. Install VLC, it has better support for video codecs. If you are getting audio only this is almost always a video codec issue. Equally so using VLC will fix it.