I will recommend you to read about PR on Google's website or do a search on the topic. Takes to long time to explain it here But anyway to give you a short advice: PR is important if you want to get your site to be included in SE FAST. Beside from that, it just a green bar. If you want your site to rank higher in SE page rank doesn't matter at all.
In theory, if I create a lot of "mini websites" on the subject of my website and place link from them to me (not the other way), will this has some kind of effect?
Google likes authority sites. Creating a bunch of sites for the purpose of linking to another site you own isn't going to give you a single authority site - and you run the risk of getting a link farm penalty. Google also checks the IP address of sites linking to you and assumes links from the same class C IP are from sites owned by the same person. They also have the ability to check ownership of the domains. Make a quality site that people will want to link to - or get links via publishing press releases, writing articles other sites can use (with a link to your site), etc. Getting a google top ranking for a competitive keyword can be a slow process.
How do you know that Google checks the ownership of the domains? Do you mean that they check the whois records?
To try and make things simple, I think the best way to think about it is that PR represents the importance of a page. As an analogy, take a medical doctor and a nuclear physicist. Both are experts and their opinions are important so they represent high PR pages. If you have a page on nuclear physics then a link from the nuclear physicist will show that your page is important and you know what you're talking about when it comes to nuclear physics (i.e., you will have a good search engine ranking because the link is relevant and important). If you get a link from the medical doctor then it shows that your site is important but it's not necessarily relevant to your topic so you may not get a good ranking even though you will have some importance (PR). Basically, PR is a measure of importance of your page and that has to do with how many other pages link to you and how important those pages are deemed to be. To be found in the search engine rankings you need both importance and relevance. You can create a site that is exclusively about one topic which means you are totally relevant but unless you get votes or links from other important sites then you won't be important so you won't be found anywhere good in the search engine rankings. This is just an analogy but hopefully it helps you understand the basics of PR and relevance as factors in the search engine rankings.
I said "the ability" as no one knows exactly what google is doing at any given moment. As a registrar, they have the ability - and they also talk about checking domain ownership and sites that are on the same class C ip in their patent. Many industry experts (including the head of Ask search) believe google no longer uses PR for SERPS. The whole idea behind PR was sites linking to a site because it had good content. The chances of an independent link from the same IP are statistically so small that google assumes that it is an owner "voting" for their own site - which is the exact opposite of what google was trying to achieve with its algorithm. If you're going to cross link sites, make sure they are on a different class C IP - which means using different hosts.