The GPL (and all open source software licenses) only covers copying, distribution, and modification of the software. In general, if you have a legally obtained copy of some piece of software installed on your computer, you as the operator of the computer have the right to use that software in any way. If you think about it for a moment, this is similar to any other non-software product that you may obtain (either for a price or gratis). For example, if you have in your possession a legally obtained kitchen knife (obtained for a price or gratis), then you also have a right to use that knife in absolutely any way. Indeed, you may even use it to perform an unthinkable action, such as to injure a person or animal. Depending on the details of such an action (was it intentional or incidental, etc.), by such a usage of your knife you have possibly broken one or more laws. But there is no need or benefit for a license on the knife to say "by purchasing or using the Deluxe Kitchen Knife (hereafter 'Knife') you agree not to injure any person or animal using the Knife."
The GPL even says so explicitly at the beginning. Of course, the GPL has no need to actually say this in order to give you this right, but the drafters of the GPL decided to say it anyway:
The act of running the Program is not restricted, and the output from the Program is covered only if its contents constitute a work based on the Program (independent of having been made by running the Program).
GPL version 2.
When do I need a license?
In general, you need a license in order to copy, modify, or distribute a piece of software. The reason for this is that these activities are normally protected by copyright law, so that's the reason you need a license (permission) to do them. If you aren't copying, modifying, or distributing the software, you have no need for a license of any kind. However, if you don't have a license, or if you were given a license but decided not to accept it (e.g. by not following its terms), but then you decide to go on to copy, modify, or distribute the software anyway, then you have possibly (depending on the details, of course) violated copyright law. That's essentially the only reason that you need the license.
If you are not going to copy, modify, or distribute software (or any other copyrighted work), you have absolutely no need for a license. This is the same as if you go to a library, take a book off the shelf, and sit down to read it. Do you need a license to read that book? What if the author wrote on the first page "This is a book about vegetarianism. By reading this book, the reader (hereafter 'You') hereby agree that You will renounce the practice of eating meat and fish and that You will eat only non-animal products hereafter." Do you need to accept this 'license' in order to read the book? No.
Note however that in practice, installing a piece of software very often involves some form of copying, such as downloading from the Internet, copying from some external media like a DVD, and so on. So it often seems like in practice you will need a license to use it, because in order to use it, you need to install it, and in order to install it, you need to perform some sort of copying. However, it is the copying that is actually permitted by the license, not the act of running the program or using it in some particular way.
I've installed the aforementioned Server and [used it] without putting my queries in public repository with copyleft note. Did I violate the GPL v2 in this case?
No. This is the normal use of the software, and you are not copying, modifying, or redistributing the software by writing queries into it. The same argument holds true no matter in what situation you use it. You could use at home, at school, as part of a non-profit organisation, as part of a for-profit organisation, as part of a military organisation, as part of a terrorist organisation, and so on. The actual actions that you perform in such an organisation is what distinguishes legal and proper behaviour from illegal or improper behaviour.