- How do we know 5800 is a correct figure if we can't even locate these manuscripts?
- How reliable is this figure?
- Which libraries in which countries in the world hold these manuscripts?
- Who first found out and established that it was 5800 copies?
Who said we can't locate these manuscripts?!
Wikipedia actually has a pretty thorough set of lists of NT manuscripts:
- List of New Testament papyri (136)
- List of New Testament uncials (322)
- List of New Testament minuscules (1–1000), (1001–2000), (2001–) (2911 total)
- List of New Testament lectionaries (2453)
The lectionaries list is the least complete here, only having a few hundred out of the 2453 it says have been categorised by the Institute for New Testament Textual Research (INTF). You could I'm sure get their full list, but you might have to pay for it, and it might be in German.
Add up the papyri, uncials, minuscules, and lectionaries and you get 5822, which is likely the source of the 5800 that you've been seeing. But any total number like this will only be a rough estimate. Our knowledge of manuscripts is constantly changing. Wikipedia notes that 47 new manuscripts were discovered in Albania in 2008, and the number can decrease as well: as scholars discover that partial manuscripts which had been categorised separately are actually parts of one manuscript.
For most scholars however the data is considered pretty reliable, as minuscules and lectionaries play a smaller role in their studies than the papyri and uncials because of their later date. Wikipedia gives a table showing the dating of the manuscripts by century. The dominant position of textual critics is that the earlier manuscripts provide stronger witnesses to the original texts. However should a new papyrus be discovered, of which we currently only know of 136, you can be sure that Biblical scholars will pay close attention to what it says.