Questions on basic ESD safely
The resistor is there mainly to protect you.  If you accidentally touch a live wire when standing on an insulating floor, you will get a very unpleasant jolt, which will probably not kill you (but don't try it!).  If you touch a live wire while one wrist is connected firmly to Earth, then a much larger current will flow, which is a lot more likely to kill you.  The resistor reduces the current to something much safer.  That said, the resistor also limits the current that flows when you first put the strap on - avoiding the static electricity jolt you sometimes get when touching metal on a cold dry day.
"Ground" can mean several things.  It can be the Earth, and the voltage it's at.  But in small DC circuits, it's often the 0V terminal of the power supply, or the negative terminal of the battery.  In the case of battery-operated gadgets, there may be no connection between "ground" and the Earth.  Here in the UK, the term "earthed" is often used instead of "grounded", which tends to be clearer.  Anti-static mats are normally "earthed", but will actually work, even if they are not.  The important thing is that you, the mat, and the circuit you are working on are all connected together.
You're trying to get rid of the tens/hundreds of thousands of volts caused by static electricity.  The odd volt here or there doesn't really matter to most integrated circuits.  ESD sensitive components are normally shipped in special packaging, which is slightly conductive.  As soon as you grab the package, the voltage equalises.
The wrist strap will discharge any initial static when you first put the strap on.  Then it will stop any new build up.