If your Quality Score is Great and Your competitor's quality score is great, in order to move higher rank do you have to bid more?
I would observe my competitor's ad text and make sure that my ad text is more enticing. That way, you can secure a higher quality score without raising your bid. CTR has more impact on quality score than minimum CPC. Note that the quality score ratings of Great, OK and Poor are very general categorisations. If two advertisers have a quality score of Great, it doesn't mean that they both have the same quality score. One of the quality scores is definitely higher. Google keeps the exact quality score hidden, so advertisers continually concentrate on improving their performances.
Yes, if all things are even (and that's difficult to measure) then bid will be the determining factor.