Is it possible to get a domain name that has a suffix of a country which is not my own country? For example, will website.co.uk become more popular in the UK? Or a website.co.ac (ac stands for Any Country) will be ok as long as its page rank is really good?

Yes, is possible. On GoDaddy you can also!

In this day and age you can purchase almost anything you'd like in regards to domains that aren't regulated by the government. Country domains such as .uk aren't on that list usually so if you need to appeal to a different market you can purchase the appropriate domain if you wish instead of a bergeric .com, .net, or .org.

Agreed. Any domain...pretty much....can be purchased, aside from things like .gov, .mil, etc, pretty much as purple said....anything regulated or owned by a government = not obtainable, otherwise have at 'er

Most can be purchased by anyone due to the automation used to register, however, though you may purchase the registration you may be required to provide proof that you meet the requirements for that countries tld. For example: .asia is coming on the market soon and it's required that the registrant be from one of the countries on that continent. Who knows how actively the registrar will enforce those rules, but the rules do exist.

There are a good number of country tld's that do not have such requirements. .cc and .bz are good examples.

Yes this is possible, but anyway you have to make sure that everything ok with rules.

Mark and Brian pretty much covered it. While you can get 'almost' any domain suffix, there are a few restrictions on some of them, so you'd need to follow those rules in order to be allowed to have those suffixes.

As for becoming popular, it really depends on the site itself, and not necessarily the suffix. A .uk is more likely to show up in UK specific searches, I think, but a similar .com address could just as easily show over there, so it really just depends.

I have several domains from different country, haven't experienced any problems either

Your explanations really did a good light on getting a good domain name. But could someone please point out to me what are the rules that I should abide in besides the geographical criteria? Anyway, who handles these domain names? Is it the government where the domain name is located?

ICANN (http://icann.com/tr/english.html) is the ultimate authority and manages all generic. See this link for FAQ's: http://icann.com/faq/#gltdrules

The country TLD's are controlled by their respective governments and are usually contracted out to a registrar.

But not all companies are ICANN-accredited. Which factors must be for that?

All registrars are accredited, with the possible exception of country TLD's. It is not a requirement for resellers of those registrars. Without accreditation registrars would not have direct access to the databases for registration.

For becoming an ICANN-accredited company has a lot of obligation you can check them from their website www.icann.org .
And for understand if the company is ICANN accredited or not check if they have ICANN logo on their website.

About extensions with foreign countries yep sure you can but it depends on which country because every country has their own rules for getting it.