Is the Work Area Bar (the gray line on the ticker along the top of the timeline) over your entire movie or only over part of it? Only the clips under the Work Area Bar will be rendered on the timeline.
Also, by default, when you've finished rendering all of the clips under that Work Area Bar, the timeline will play, which is what you're seeing. (You can change this in the preferences.)
I believe a fundamental principle is being overlooked here since you report
The first clip of the project has the green "rendered" line, whereas the rest of the project has no line above the clip.
A green line or no colored line over the Timeline content is the project telling you that you have the best possible preview. Consequently, no Timeline rendering is needed nor available. When you press Enter to render the Timeline content in that situation, all you are going to get is Timeline content playback and no Timeline render.
On the other hand, if you saying that you have an orange line over any of that content, that is the project telling you that this Timeline content needs Timeline rendering to get the best possible preview. Then Timeline rendering is available to you when you press the Enter key or one of the other commands to Timeline render.
When you see no colored line over content that typically means that you have imported source media with the right project preset. The no colored line will remain until you edit that content. Then you will see the orange line which will turn green when you Timeline render that content. That content will never return to the no color line. Subsequent edits of it will give orange line over content. And Timeline rendering of that will give green line over content and so on.
Please consider. Any questions or need clarification about the above, please do not hesitate to ask.