Does density altitude affect actual altitude?
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The basics


Height: absolute height above ground, measured using a radar altimeter (like the helicopter on the right)
QFE: the barometric setting an airport broadcasts that makes an altimeter on ground read zero. This gives height only around the airport if the area is relatively flat. Also known as Above Field Elevation
QNH: the barometric setting an airport broadcasts that makes an altimeter on ground read the airport's elevation (height above mean sea level)
Aircraft using the same QNH below the transition level* in the same area are all flying the same indicated altitudes
Aircraft using QNH 1013 mb (29.92 inHg) are all flying the same pressure altitudes. Above the transition altitude* they're called flight levels


* The transition altitudes and levels vary from place to place. For flying below 18,000 feet in the US, you can ignore flight levels for now.

QNH is good for separation since everyone is using the same measurement system. Even if the temperature's rate of change with altitude (lapse rate) is different from the standard.



Lapse rate


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On the left is a hotter than usual lapse rate, i.e., it gets colder less slowly. Still, all aircraft using the same pressure setting will still be flying the same indicated altitudes / flight levels.


Pressure altitude (what the altimeter reads when set to 1013/29.92) adjusted for lapse rate deviation is density altitude. It's used for performance calculations.


The answer

You said, "[The] atmosphere actually expands and becomes taller under hot conditions".


Yes, but not for that reason. A hotter than standard day means the air molecules are more jumpy, i.e., they are producing more pressure. That's why 1,000 feet on the left is higher. Think of it this way, air is hot, it's still keeping pressure a bit higher (height).


Remember, a basic altimeter senses only pressure from the static port, it doesn't know the temperature.


Since the pressure is the same for the same altitudes, and the variable is the temperature, it will be lower air density on the left due to the higher temperature (angry molecules don't like to cozy up). That's why engine output and lift are impacted.


For cold weather, usually at or below 0°C (32°F), this deviation can become dangerous in mountainous areas. There are special rules and equipment for that, which are listed on the appropriate charts for airports that usually face such conditions.



Further reading:


Wikipedia: Flight level
SKYbrary: Altimeter Temperature Error Correction
SKYbrary: Transition Altitude/Level
How is non-standard OAT accounted for?
How does altimeter reading vary when departing from ISA conditions?
When calculating takeoff performance why include airport elevation in pressure altitude?
Why is Cessna 172 POH performance chart based on Pressure Altitude and Temperature instead of Density Altitude?