It would make sense for do-follow links to improve the speed at which a site is indexed, since they do give the indication that relevancy should be added to the link
Google crawling is not linked with dofollow links.Yes when you have doffollow link google crawls it and when it is nofollow it doesn't crawl it. So dofollow links always crawled. But it doesn't effect the whole sites crawling rate. Crawling is depends on how frequently the content is updated. Is the content original? Is the content interesting enough to stay users?
IMO, no matter whether a link is Do Follow or No Follow when it comes for Google bot crawl but having a Do follow link will certainly help you in SEO. If you manage to add more and more content to your website then Google bot may keep a close eye on your website for any updates !
Dofollow links will not increase the speed of Google Crawl. But dofollow links are good for seo and also to increase the page rank of your inner pages.
Yes off-course Google gives much importance to Dofollow links than nofollow links. I follow 3 methods to improve ranking to my website such as Blog posting, Forum posting, Article submission. I have got good result by  following these steps.