If you're serious about your IM career, then I believe it's a MUST to pay for a domain. Because simply, with a free domain you risk all of your Search Engine Optimization efforts. A while ago, Google deindexed all of .co.cc domains, so imagine that you work on your site for a considerable amount of time and wake up one day just to find out that you've been deindexed for some reason. Additionally, they look spammy and unprofessional.
Regarding your second question, it's not about the number of backlinks. It's not a numbers game - ask anyone who've been in this game for some time and he's going to tell you exactly this. Getting a few thousand links in a matter of days or even hours is VERY doable using automation tools. But these links are usually of a low quality. What Google pays attention to is the authority of the page that's linking to your page/site + domain age/Google Page Ranking/number of outbound links/link age, etc.
So pay more attention to link QUALITY - but at the end of the day, the more links you build the better chance of getting more link juice to your site/page. I personally have PR3+ sites that are ranking top 3 for medium competition terms with less than 100 links total.