Are graduate-level courses weighted higher?
Relative to your core undergraduate courses, I'd imagine so. Taking advanced graduate courses (preferably, more advanced versions of your core classes, since this will look like an upward trend if you didn't so hot in your core ones), and doing well in them, implies one can do potentially good research in grad school. Courses at this level will tend to expose you to current research that you may work on in the future. Aside from LOR and conference/journal publications, this would be the next best indicator of your potential success in grad school. Which leads to the next question. . .
Is this a significant part of the decision process?
I wouldn't say that it's significant (consider that some liberal art colleges may not have available graduate courses to undergraduates, as compared to a research university) but it's still important, if the resources are there. Graduate admissions are aware of that, but they key is to stretch your intellectual vitality as far as possible. Remember that there aren't too many indicators of whether or not an undergraduate can do well in grad school (after all, the "research frontier", depending on the discipline, may simply be inaccessible, even to an exceptional undergraduate). This applies to other fields like mathematics, but three factors will always be taken into account by any respectable institution (LOR, statement of purpose, research experience), and you've already mentioned them.
Aside from that (and GPA, which is really more of a "cut-off" tool), if all else is taken into account, it will probably come down to best fit; whether or not your research interests coincide with the faculty of your chosen area. If you wanted to do research in artificial intelligence, it wouldn't make sense to apply to a program that (hypothetically) only specializes in mathematical optimization, so you would probably be passed over to someone who does want to do research in AI.
aeismail had a nice answer so I will not add much more regarding "red flags" (I would say that even if you did terrible, say, your freshman year, the best thing to do would be to show an upward trend in your grades). However, remember that the main question admission committees want to answer is "Can this person succeed in our program?" (i.e can this person do research?), so you have to persuade them to say yes. Your application will be scrutinized by professors who want to see if you're worthy. Coursework is important, but demonstrating your research potential in ways that are not just related to coursework is essential.
The best thing to do would be to create a check-list of what they expect to see in your application, and meet all of the requirements, then go above the call of duty (whether it's in the form of a publishable senior thesis, which I would argue is always better than chugging coursework, or presenting your work at a conference, or getting an awesome research-based LOR from a professor who knows you well and can vouch for your potential, or all of the above).