In which "order" does Windows manage RAM?

I'm asking this question because I was having memory problems (BSODs, MemTest errors), which seem to have stopped after I switched the faulty stick from SLOT3 to SLOT4 (as labeled in my motherboard).

Does Windows tend to use the SLOT1 first, and then SLOT2, SLOT3, SLOT4, etc. according to how the motherboard decides to map the memory? For example, if I have 4 sticks of 2GB each, and I'm not using more than 2GB of RAM at a given moment (say I'm just staring at my desktop, or listening to music), do the other 3 sticks remain inactive while Windows only uses the first stick (SLOT1)?

It doesn't work that way. The OS doesn't even know about these numbers.

The motherboard (bios) organizes the strips into 1 or more blocks of memory that are handed over to the OS. The ordering isn't necessarily in the sequence of the slot-numbers.. It is possible to SLOT4 strip of RAM will actually hold address 0 and that the highest address is at the end of the strip in SLOT2.
Also (multi-channel RAM) it is possible that slots are used in pair (or even triplets) with them memory used interleaved: odd memory rows coming from SLOT1 and even rows from SLOT2.

And to make it even more complicated. Any modern OS uses virtual memory. How this virtual memory space is allocated on top of the physical memory is mostly handled by the MMU (Memory Management Unit) which is part of the hardware (CPU and/or Northbridge chip depending on which particular CPU we are talking about). How the MMU shuffles memory around to present it to the OS can vary a lot.

To further complicate matters Windows will spread its usage over the entire RAM space. Uses the lowest part for the OS and interrupts, but drivers and other hardware related stuff (like video-buffers) typically get allocated at the end of the RAM space with programs somewhere in the middle.

TL:DR: There is no general rule determining how a RAM slot eventually gets used by the OS. The numbers on the motherboard are just there to tell the user in what order the SLOTS must be filled. They don't mean anything to the OS.

On any modern OS (not just Windows) you can't have the OS avoid part of the physical RAM in order to work around the bad spot. (Like physical hard-disks do by mapping out the bad blocks.) You will have to physically remove the bad RAM stick or avoid putting a good stick in a bad slot.
And working out which stick or slot is the bad one can be a major problem as you can't really tell from the outside if you have a bad stick or a bad slot.
The only way to test that is to try the sticks, one at a time, in a known good motherboard.

Most likely the act of removing and reinserting the DIMM fixed the problem, not the swap of places.

DIMM contacts carry frequencies in the GHz range, where really strange things happen. It is not random that contacts are often gold plated. "Reseating" a DIMM will move the point of contact by fractions of a millimeter and may easily move a contact's quality from "just not good enough" to "just good enough"