if my brain forms a new connection (register a new information) it can't really know "where does the new element stand in the system of things"
There is a lot of contention on the specifics, what we have are theories of Memory and Information processing in the brain, and as such nothing I am going to mention is settled.
In a nutshell, there are 2 main overarching themes into how the brain processes environmental information: Maps and Hierarchies, perhaps the most researched map is that for vision or what is know as a Retinotopic map this is a simplification,there is a lot of processing and multiple overlapping maps ( movement,orientations,color),1
the Hierarchy part comes into play as the outputs of these primary maps combine to make more novel maps like edges and shapes, other parts of the brain in turn connect these primary and secondary maps into higher meaningful percepts, perhaps the most striking feature, is that you recreate the original pattern as you remember, that is the neurons (or a subset) that fired when you first experienced something, fire again when you remember it for a second time2.
This scheme more or less repeats through the brain with a number of specializations3, we devote a lot of real state to vision,social behavior ( maps and hierarchies of how to behave in public and what other people are thinking), language ( maps and hierarchies of words and meaning or a mental lexicon), as well as personal history (all the people that have wronged me, my favorite food), and finally, we have areas devoted solely to the association of other areas, a map of maps and a hierarchy of hierarchies with temporal buffers.
This was a whirlwind overview of how we believe information is organized, but it does not answer your question since I have mostly neglected mentioning how the information is stored and retrieved, that is mostly the domain of memory.
Memory is not a unitary concept and by that I mean that there are a lot of behaviors that are bunched all together into the word memory ( there are around 11 different types for instance)4, I am skipping much of the details that are not relevant, but we believe that the following mechanism is at play for the type of memory you are describing:
When you experience something, besides direct access to your primary processing maps and hierarchies, an extra or a number of extra copies are generated and routed through a switchboard, ( the Hippocampus and related structures), it is in this switchboard that the interconnections necessary for associations are first generated, and later through a process of selective repetition, the relevant ones are preserved and eventually consolidated closer to your primary maps and hierarchies.5
Is it possible that the brain somehow brings the knowledge up when I'll hear about it in school ten years later?
Yes, the specific phenomenon is called priming, granted, in light of the previously discussed models, It will depend on some variables, was the information stored originally in long term memory, has there been interference from other more recent memories, how much time has it been.
It seems weird to me that the brain would automatically find the exact same neurons where the information had been stored before.
Yes, it is amazing, the information finding happens due to the organizational scheme previously described and another phenomenon called plasticity or synaptic plasticity, this last part utilizes the connective weights in between neurons to both preserve ( connections with higher weights get preserved) and then finding related connections ( through a path of least resistance in between neurons with high weights). Additionally, long term memories utilize other strategies and optimizations and there could be further temporal and functional aspects to encoding,storing and retrieving. 6
In other words: Can the totally forgotten information still be useful?
Yes,provided the information was encoded and stored originally,even after years of non use, there is evidence that a priming effect might still be available.
Notes, reference & sources:
1. For an early look at Hierarchies and maps I think Vision (Marr) is a good introduction, Vision science (Palmer) has the full overview.
2. This one fact deserves more attention, a good overview is found in Cognitive Science (Gazzaniga)
3. Neuroscience (Purves) & Cognitive Science (Gazzaniga) both deal with how the neocortex subdivides and specializes.
4. Cognitive Science (Gazzaniga) has a great overview of Memory and The Oxford Handbook of Memory (Tulving,Craik) is a comprehensive bible of sorts if slightly outdated.
5. Gateway to memory (Gluck, Myers) has a very readable and detailed overview of the hippocampus, Rythms of the Brain(Buzsaki) expands it.
6. In search of Memory (Kandel) deals with plasticity at length.