Disclaimer: I am a current doctoral candidate with 2 years of experience as the graduate student representative on the admissions committee of my department (in STEM) at a major R1 university in the USA and so, write this answer from that context.
Simply put, a strong research record matters. However, this comes with several caveats. For a successful doctoral application in our department, we look at
1. Standardized Test Scores: These (GRE/TOEFL) will not get you in but can keep you out. Mostly, they serve as preliminary filters and to fulfill certain graduate school minimum requirements.
2. GPA/Academic Record: This matters significantly. Relevant grades in relevant courses matter more. A 4.3 GPA from MIT in Mathematics and Computer Science is great for a Mathematics or Computer Science application. We tend to look at overall GPA's and then at relevant courses to catch red flags. For instance, you want to develop novel machine learning algorithms but have a C+ on "Introduction to Discrete Structures" and B- on "Statistical Data Mining". Red flag !
3. Research Experience: Research experience is important but more so in different sub-fields. This is the same as publications but there are important variations. For instance, in the field of Human-Computer Interaction (HCI), having a first author paper in CHI or TOCHI matters more than a first author paper in a conference the admissions committee most probably haven't heard about. If you want your research to speak up, then get published in the best conferences in your field. This is not a requirement but is a very good sign of your nascent research potential. Over the past few years, I have seen increasing numbers of applications to our department come from well qualified undergraduate and masters students with such records. Its a competitive world !
4. Letters of Recommendations: This should be correlated positively with research experience. If your letters talk about your wonderful performance in coursework, admissions committees are generally not very interested (they can already see that from your transcripts). If, on the other hand, you have detailed letters of recommendation from researchers that the admissions committee might know and which attest to your research performance with them, then that is a very good sign indeed !
In other words, its all about signals, signs and red flags. Specifically, in response to your questions:
Unpublished manuscripts are not worth much but can serve as a good writing sample in case you do not have any other publications.