Preview Keeps Stopping Once Rendering Is Done In After Effects

It looks like you are using After Effects as a video editing app. That's not what it was designed to do.

When the Ram preview reaches about 8 seconds it looks like your resources are filled up. The green line in the timeline is showing more in cache with holes and until those holes are rendered you may have problems getting a preview to start again.  I don't see much that isn't normal behavior in your project.

It does look a little odd when the preview hangs. Your Preview Panel is set to Work Area Extended by Current Time, but the work area appears to start at the first frame of the comp and extend beyond the visible end of the timeline. Try changing that option Work Area or Play Around. Try turning off Cache Before Playback. Something might be hanging up.

I always try and limit my comps to a single shot if I can. If I need to create a sequence in AE I limit that to a sentence or phrase. If it's a music video, it's a measure musical phrase, then I do the editing in an NLE like Premiere Pro. You end up with a lot more comps in a project but it's a lot more efficient in the long run. I'm currently working on a project that needs about 20 seconds of audio covered by things I can only do in AE. My longest comp is 4 seconds. I have 9 of them to cover the required edited sequence. I wouldn't think of trying to pull off all 20 seconds in a single comp because that comp would have more than 100 layers and by the time I got into it about more than 10 seconds the ability to fine-tune and edit would become incredibly frustrating and slow. You might consider breaking up your comp into much smaller sections.