Yes very much so for sites linking to you dofollow are the best i am also looking in to this, if your site is dofollow it may get targeted for spam i believe
Dofollow affect raning. But nofollow may be good for bringing traffic. So don't get discouraged just because a site is nofollow
Dofollow are the links that google tracks and play a huge role in SEO. However, traffic plays a large role in your ranking as well and many nofollow links can drive lots of traffic. You should have a healthy mix of the two
certainly, there is lot of difference in both link types..... DoFollow link can be read by search engine crawler where as NoFollow links are not readable... and can not count them as backlinks .. thou, it can turn some traffic back to your site but other than this NoFollow links are good for nothing ....
It may be that the nofollow is also affected in some ways, at least the links appear in Google Webmaster Tools.
You need a mixture, definitely. Followed links allocate Page Rank to your site, which help promote your site within the search engines. However if you build an unnatural number of followed links from an unnatural distribution of sources, you're going to be seen as spammy, so they need to be balanced with nofollow links too. Anchor text is also important - keep it relevant to your site, but varied and natural looking - too many links with the same anchor can be very dangerous.
No - follow is just a webmaster saying that they cannot vouch for the link to your site. Basically, it is a spam reduction tactic. Be sure to have a balanced mix as, although no-follow does flow pagerank, it also brings natural traffic (which is what you're trying to gain from SEO anyway), and it helps to prevent google flagging you for spammy link generation.
Links that are "follow" affect your ranking because you are telling the search engine to follow that link. "nofollow" links are still good because users could still click on them to get to your website but search engine won't be able to access them because you're telling them not to follow them.