There are so many questions in your post, some asked, some unasked. So you are not going to get everything you need to know in a short summary.
However, let's go with the basics.
A voltage when applied across a resistor produces a current through that resistor. The current is proportional to the voltage and inversely proportional to the resistance. If you double the voltage you double the current. If you double the resistance you halve the current.
Now, to divide voltages, you need more than one resistor.
Put two equal resistors in series and apply a voltage to the outer ends. A current will flow through the resistors, the same current through both resistors. Since both resistors are equal and the current is the same, the voltage across each resistor will be the same as the other. And these two voltages must add up to the total voltage. So by measuring at the half-way point between the two resistors, you have divided the applied voltage by two.
Now, it's not quite that simple, however. Because if you try to draw any current from the half-way point you will upset the balance of equal currents and now the voltage will move. But as long as you only try to draw a minute current (such as with a voltmeter) the divider will be fine.
Lastly, you need to know that a "9 V" battery is not actually 9 V exactly. It is approximately 9 V, give or take a volt or three up or down (one up, three down). So you can't get exactly 6 V using the voltage divider method unless you first make sure 9 V is exactly 9 V.