I'm a PhD-student in the beginning of my second year, here are my answers to some of your questions from my (limited) point of view.
First of all:
Another point in favour of pursuing this path is that I'd like to have a more integrated view of biology (and adress my studies in such a way), maybe in despite of a more detailed and defined one.
Now that I'm working in bioinformatics I feel that I easily start to drift into a very "un-integrated", highly specialized view of biology - that is, no biology at all! I easily focus completely on algorithms, on how to clean data, on how to get it to play along nicely, on finding a pretty way to plot certain behaviour! That comes back to bite me when I do talks or have meetings about my results, I easily "forget the biology behind it all", as my advisors say. That is something you should keep in mind if you focus on this career!
Would I stop definetely working in the lab? Bioinformaticians do run their own esperiments or they spend all of their time working on the computer, analizying other scientists datas?
Most bioinformatics researchers I know do not work in the lab at all. Why would you? All that pesky security-training, just to stand around dangerous chemicals and perform the same movement 500 times, as a bioinformatician you can have a nice hot cup of coffee and a cozy chair!
About other people's data: Yes, you analyze their data, or how I like to see it: I tell them using their "old" data which new data to generate and then I have a look at it. This way I don't have to sequence stuff in the lab but still get the benefits. I definitely run my own experiments: for example, we have mountains of genomic reads that were used in assembling genomes, but you can do so much more with! (call SNPs, call indels, do maybe even population studies...) The few computational biologists I know work with theoretical models so they don't need other peoples' data.
What are the prospects (scientific challenges and work possibilities) in the future for a student choosing to study bioinformatics now? Wouldn't I risk to become an useless "hybrid"?
I have no idea. I do know that getting a job in the field right now is much, much easier than, say, as an ecology PhD. All the former PhDs I know in my lab got a job offer during the second year and were never unemployed.
It's hard to tell what the scientific challenges are going to be - right now, bioinformatics as a field is struggling with reproducibility of studies, as most published papers don't really include the necessary details you need to reproduce them. The discussion is ongoing and will probably lead to a slightly different model of publishing, i.e., more focus on arXiv, more open data like with GigaScience, journals that force you to publish your code with your paper, more open-ness in general, maybe even forced tests (Haha as if)?
The specific challenges per sub-field are diverse, what do you want to focus on? If you focus on plant-genomics (my field), we still don't have assembly-algorithms that properly handle polyploidy, or whole genome duplication-events; if you focus on genome assembly, the next generation of sequencing machines will have a much, much longer read-length, and none of the current algorithms really use that information (maybe ALLPATHS?). Plus PacBio-reads seem to be dirty all over the reads, and the current cleaning programs are very slow (at least for my plant datasets). There are many other sub-fields like evolutionary bioinformatics, no clue what the problems there are.