Most major topics have been covered in discussion, so just two remarks/experiences:
While director of graduate studies at Northwestern (2007-2010), I led a committee which valued undergraduate preparedness over research experience. So at least as far as Northwestern was concerned during that time frame, research (especially research for which an undergrad may not be fully prepared) did not help as much as you might have thought.
In trying to use RTG funds toward undergraduates, rather than try to simulate a research environment, I and my co-PI's created an "undergraduate conference" to try to offer a supplement to standard undergraduate curricula, without yet getting on toward research. Here is the link: http://www.math.northwestern.edu/summerconference/ Maybe we'll do it again next year?