Is it possible for a high school student with no academic qualifications to publish a research paper? I am a bachelor student at the second year of studies (Physics), but I'm an independent programmer since many more years. 

Now, I have some data about an algorithm I've done and I would like to present the results on a research paper - I have a high level of understanding of the theory behind, and some background of theoretical computer science. Is it "reasonable" to write a paper with no titles at all (high school) and without the name of my university maybe? 
I don't know whether include it or not, since the topic is not related to what I study there.
You should definitely write it up. There are plenty of good reasons for doing so and no downside. Writing a paper is a good exercise for future paper and report writing, and while you are doing it you are likely to think of new ideas about your research. You will also have to search for similar work that might be suitable for references and you may find something useful there too.

You should also put the paper online somewhere so that it is preserved and available to others who may find it useful. If there is data and/or code that goes with the paper you can use figshare for example. It does not matter if there is not much response to your work, that is common when authors have no background and does not mean it is not good. Someone may pick up on it later when you don't expect it.

You might also want to try to publish in a peer-review journal if you think it is sufficiently original and useful. That might require you to write it up in a specific format such at LaTeX. Again that is all good experience. You will have a much better chance of acceptance in a journal if you can use your university affiliation but you will have to make it clear that you are an undergraduate there to avoid problems and it might be best to check with your tutor. If you search around you may find a suitable journal that is specifically set up to accept work from undergraduates. If you can get it accepted it would look good on your CV whatever you plan to do in the future.

The only circumstance that would make it unwise to publish your work would be if the algorithm is original and of high commercial value and/or patentable in which case you would have to follow a different route to protect your intellectual property. However that is a rare situation.