One more thing, and I guess it slightly contradicts what I
described before. There is one outlet (#1) in the room that is
not controlled by the light switch(#2). That outlet is closest to
the circuit breaker box. As you go clockwise around the room, the
next thing you reach, on the same wall, is the single light
switch(#2). Around the corner and on the next wall is the duplex
receptacle (#3) with two whites, one black, and an unconnected red
wire. The next wall has two more outlets (#4 & #5), and both work
only when the switch is on.

#1 is always hot. #2 has the switch with a wire that is always
hot. 3, 4 and 5 are switched. Is the mystery red wire a way to
get the last three outlets hot all the time? Or should I stick to
my day job?

Phil