A1. Repeating keywords on a page has diminishing returns; however, if you aren't using the keyword you can't expect to rank for it. In some cases one exact match keyword may be enough if you are using synonyms throughout the content. The number of times you should use a keyword also depends on how long the copy of hte page is - the longer the page the more keywords you can incorporate without it sounding unnatural.
I always try and use a keyword in the first sentence of a page, and the last sentence of a page. If you can naturally fit it into the content a few more times it will not hurt you. I'm assuming SEOmoz grading is based on correlation studies. They must have seen 4 keyword mentions correlates with higher rankings. Check out this video: http://www.youtube.com/watch?v=zHYA5gAsRBA
A2. Stick to one H1 tag. I like to think of it in a couple ways:
The H1 tag may not even be a ranking signal, but rather that H1 tags are traditionally used at the top of a page. Is the H1 tag the ranking signal or simply mentioning the keyword at the top of a page the ranking signal? Having more than 1 H1 tag is unnecessary and may just send a red flag to Google that your are over optimising.
If the H1 tag IS a viable ranking signal, then the more you use it on a page the less effective it may be. If I have a paragraph and I bold every other word and ask someone to pick out the most important word it is going to be very difficult. If I have a paragraph and only bold one word and ask someone to pick out the most important word it is very easy.
A3. Not really sure what you are referring to, but SEOmoz does have its own software to crawl your site and gather the information presented in your Moz campaign. Additionally, the information in OpenSiteExplorer is their own.
A4. Rankings and traffic are never, and should never be, guaranteed in SEO. However, if you are paying attention to these elements and optimizing properly I'm sure you will see some results.