Yes, you can get pr for your landing page you are promoting one. Try to get niche links with relevant and pr.
Yes, page rank is pagerank,not site rank. And you increase it one way and one way only - backlinks. (of course, the next question is why do you want to incease it)
promote landing page and get more backlinks for that page but you should keep in mind relevency does matter.
digg and stumble dont pass like juice. Write articles and sumbit them to various article directories.
Digg and stumble has nofollow tag. Check it out on digg and stumbleupon. You better try to get traffic from them rather than increasing serp. If your traffic will increase ultimately your SERP will increase. Increase in traffic allows visitors to add to you if they like and ultimately your serp would increase.
Better concentrate on increasing rankings rather than page rank. Page rank is not related to actual SERPS ( search engine rankings) of your website in any way. It is just a tool to gauge how old a website is ? Other than that, it has no significance