Let's say that I want to keep meat in my fridge for a week. From what I read, this is frowned upon, as it leaves time for bacteria to develop and to produce toxins which won't be destroyed by cooking the meat. However, I also read that in cold, salty conditions, only favourable bacteria develop, and that's how sauerkraut and other pickles are made.  So, can I safely keep raw meat in the fridge for a week (or more) if submerged in salty water ? What would be the appropriate salt concentration and maximum keeping time ? 
What you are essentially describing is a brine. That is a saline water solution that trough the process of osmosis draws in moisture trough the cell walls and releases them. This has the effect of making the cell walls more absorbent of moisture which leads to juicier meat.  The problem is for brining you typically want to leave the meat in the brine at room temperature. Doing the brine at cold temperatures retards the osmosis effect. Now that does not mean it cannot be done, it is just going to take 2 to 3 times as long.   As for the concentration of salt that depends on what you are brining and what level of saltiness you want. For veggy pickles I have seen solution as low as 5% but for meat you can have a general rule of 6% - 8%. This means 60 - 80 grams of salt per liter of water.(Weigh the salt, different salts have different volumes.).  As for how long brines can last, if you take accounts of how the sea voyages that explored the new world seriously then brines can keep meat for up to two years.  If you live in the new world it is fun to think about how different our countries histories would have been if the European settlers could feed themselves in their long sea voyages with brined meat. It is actually the food stuff that made their voyages possible.     However, I also read that in cold, salty conditions, only favourable   bacteria develop, and that's how sauerkraut and other pickles are   made.   There are four main types of lactic acid bacteria are commonly present in sauerkraut: Leuconostoc mesenteroides, Lactobacillus brevis, Pediococcus pentosaceus and Lactobacillus plantarum.     Sauerkraut is made by a process of pickling called lactic acid   fermentation that is analogous to how traditional (not heat-treated)   pickled cucumbers and kimchi are made. The cabbage is finely shredded,   layered with salt, and left to ferment.   SOURCE 