How to calculate heatsink requirements? Explanation of the K/W unit
Just for reference, I'm going to use °C/W because it's what I'm used to.

In the datasheet for your regulator under the thermal characteristics there are typically two values.
\$R_{\theta JA}\$ - Junction to ambient
\$R_{\theta JC}\$ - Junction to case

If not using a heatsink then the \$R_{\theta JA}\$ is the value you use to calculate how warm the regulator is going to be.
When using a heatsink you take your \$R_{\theta JC}\$ and add this to your heatsink rating (20°C/W in this case) also remembering to add in any other thermal resistances. The thermal pad between the device and the heatsink is one such example.

Now this 22°C/W is in a perfect world where it has a super perfect thermal connection so take your result as an estimate rather than a "It will always be 60°C"

When you calculate the temperature, you are correct in thinking this will be the temperature of both the component and heatsink. However, remember that there's a lot of difference between a TO-220 package and a block of aluminium and it will take time for both items to reach the same temperature.

Edit: Because I remembered I had it to hand, here's a thermal image of a PCB I'm working on. This is about 5 minutes after switch on, notice the component itself is 43°C and yellow on the thermal image but the surrounding large copper heatsink pad for the device is a purple and closer to 20°C.

P.S - Please ignore the white hot supernova approaching from the right, this is a work in progress board