Pagerank, the rank (0-10) given by the Google to evaluate the importance of the webpage and to rank in their search results. After the lastest PR update some website gaining and some are dropped their rankings.
Important but not as important as other factor. Such as the quality of the content and the social engagement of the website
Page rank banklinks influence your serp and page rank influences your title tag. Content is the most important factor on your website with the newest google algo, but content without page rank. Is like a car without an engine, it's possible to move it further. But the effort required to do so is much harder.
Page rank only tells us the rank of your incoming links. If you have a few PR4 and above links you will have a high PR but it does not necessarily translate into higher SERPs. Higher SERPs is more important than higher PR.
The answer can be "YES" also, and "NO" also. Why YES : 1) PR is still one of the 200 odds ranking signal Google uses. 2) If you get high PR, people will want to link you often. 3) High PR still matter if they are relevant and natural. Why NO : 1) The calculation of PR is biased in some of the cases.
Only your 2) YES reason is correct here. And its basically the only significant one in favor of the PageRank. Reason 1) is actually more in favor of NO as this is exactly why PageRank is not so useful - its just 1 among 200+ factors. I'd say that PageRank's value decreased a LOT in the last year or so. Google stopped updating it regularly, the last major PageRank update was not even planned by them. Matt Cutts actually hinted a couple of times that Google themselves are aware how biased it might be and they work towards creating other metrics while PageRank will slowly continue to be devalued to the point of being non-existent in the future. Forget about PageRank, Alexa Rank and similar metrics, only people with outdated knowledge still continue to praise them and use them as a bait to fish clients with no idea of SEO. Just focus on your relevant and quality content and backlinks and your traffic will increase naturally. After all your end goal is to get more people to find and view your website, not just boast with your high ranks and have a generic website that will eventually achieve a very high bounce rate.
Page rank is important of course, Because It shows many things for the website, like if site has done good link building or not? Is site has good amount of quality contents or not? and site's page rank shows it's online authority and reputation as well.
On the contrary I can show you some websites with PR 4-5-6 which are comprised of Black Hat link building and content which is hardly readable. Yet Google has not penalized them so far What about them? How are they showing quality link building or content?
Page Rank is very important but it isn't all. What do you think if your site has a high PR but it doesn't appear on the top of search results? Thus, having backlinks from high PR sites is good but you still need to optimize your page in order to rank high in the search engines.