How is pressure related to air density?
The atmosphere approximates an ideal gas, and as such you can relate pressure and density through the ideal gas equation.  The form we use in meteorology uses mass density and is given by:

$$p={\rho}RT$$

where $P$ is pressure in units of Pa, $\rho$ is density in units of kg m-3, $R$ is the gas constant for dry air (287 J kg-1 K-1) and $T$ is temperature in Kelvin.  This assumes a dry atmosphere and humidity will decrease density for a given pressure.  Consideration for water vapor is usually brought in by changing temperature into virtual temperature $T_V$ where $T_V=T(1+0.61q)$ and $q$ is the mixing ratio of water vapor (units $kg~ kg^{-1}$). 



Pressure increases with temperature because the particles have more kinetic energy (which is proportional to $T$).  Imagine a box full of bouncing balls, if these balls start moving faster the balls will hit the walls of the box harder, imparting more force on the box.  Pressure is merely force per area, so if the force increases but the box stays the same size, the pressure has increased.



Air density can decrease with temperature if pressure is also decreasing.  If pressure is constant, this cannot happen (they would be inversely related).  Any time you specify a relation between any two of pressure, density or temperature you must hold the third constant or specify its behavior.

For example, hot air rises, but why then is it cold on top of a mountain.  The answer is that hot air is less dense than the cold air surrounding it for a constant pressure, and being less dense it rises.  With a mountain, the pressure is decreasing, and we likewise find in the atmosphere that temperature decreases with decreasing pressure.



On a hot day what tends to happen is that the surface, which is being warmed by the sun, heats the lowest level of the atmosphere, reducing its density (it is at the same pressure as its surroundings and its T rises).  This will eventually drive convection and mix this warmer air vertically.  Given enough time, this will reduce the mass in the column of air and therefore reduce the pressure at the surface.  These are called "heat lows" and you can see them forming in the desert areas and they play roles in sea breeze formation and the monsoons.



To address the expanded question:

The point in the FAA written is best understood by forgetting that we fly at constant altitudes -- we don't.  In level flight we fly on constant pressure surfaces which we then translate to an altitude.  In any given column of atmosphere, if it is warmer than standard a given pressure surface will be higher and when colder than standard the pressure surface will be lower. 

To illustrate, let's consider you are flying at 3000 ft or roughly 900 mb.  Everywhere on this pressure surface will indicate 3000 ft on our altimeter for its current setting.  If we go somewhere hot, this pressure surface rises, and so we climb (though we think we are level) with this pressure surface but because the pressure has not changed, we still indicate 3000 ft.  However, we are higher than 3000 ft in reality.  

This follows into your next question.  Aneroid wafers detect pressure changes and your altimeter displays an altitude not corrected for temperature.  This is why your true altitude can vary with temperature for a constant indicated altitude.  When you correct the altitude for temperature we call this "density altitude".  

So back to my example above, your are flying along at 900 mb and indicating 3000 ft, and heading into warmer air.  The pressure surface starts to gently rise and as it does you are not yet following that rise and your altimeter will indicate a descent.  In true level flight you will begin to fly into higher pressure in this case as the 900 mb surface rises above you and the aneroid wafer in your altimeter will indicate a lower altitude and a descent.  You correct this and climb back up to the 900 mb pressure level so that your altimeter will once again indicate 3000', all the while actually gently climbing on this pressure surface.  You won't really be cognizant of this while flying however, and will just minimize vertical speed and maintain altitude blissfully unaware that you are really flying on a sloping constant pressure surface.

To better illustrate this, consider the following figure:


In this figure the reds signify a warmer than average column of air and the blues a cooler then average column.  The whitish area in the middle is a column at average temperatures.  The black solid lines are isobars (lines of constant pressure).  The dashed black line is a true altitude above the surface. Finally, the bold black line is the pressure level that corresponds to the true altitude of the dashed line at ISA conditions.

What you should notice is that the pressure levels in the warm column are spaced further apart because the air is less dense and more of it is needed to produce the same pressure (as pressure is just the weight of all the air above it).  Likewise in the cool column the pressure levels are spaced closer together because the air is more dense than standard.  

To tie this into the discussions above, consider yourself in the standard column (white background) at the true altitude above ground represented by the dashed line.  Your altimeter does not sense this true altitude but instead senses the pressure outside of the airplane.  This will be roughly calibrated to your true altitude (uncorrected for temperature) but using the local altimeter setting.  Now as you fly either to the left or to the right and maintain a constant indicated altitude, you will track along the bold line, as this is the pressure that corresponds to your true altitude at standard temps.  As you fly toward a colder column you will in reality descend, and you will climb as you fly into the warmer column.