How long is the period for google to trust a completely new URL?
Trust is earned over time with links. Some sites will gain it very quickly, while others will never achieve high levels of trust.
The first question is, how exactly do you define "trust"? You could use PR to measure trust, but everything is relative. If you only consider a PR 10 site as trustworthy, it is 99.99% likely that your site will never be trustworthy. As of Aug 4th, 2011 there are only 14 PR 10 websites (pages) worldwide, yet there are tens of millions of websites.
Even if you establish a certain level as trustworthy, such as PR 7, the next issue is measuring PR. Google only releases PR toolbar updates 3-4 times each year, but the figures are updated daily internally.
If you decided PR 7 was your goal (as an example) then it is possible to achieve a PR 7 site quickly if you could pump enough resources (i.e. money) into the site. If you created a well-designed, quality site which offered a product, service or information that was credible enough to cause enough interest, then it can certainly be done.
do you think it is more important for a site to get say, 20 backlinks from anywhere. Or 5 backlinks from very trusted blogs, etc.?
I would prefer 1 quality link in content from a trusted blog or other quality source then 100 "backlinks from anywhere".