After swatching and blocking, is it common to unravel the swatch and use that yarn in your project? I'm assuming the blocking would change the yarn a bit but that once you blocked the whole finished piece it wouldn't matter?
You could, if running out of yarn and the small amount would save you.
But after unraveling you should steam (or wash if the yarn does not change by that) to relax the yarn, otherwise you may get a different look as the yarn is no longer straight. (Thanks fixer 1234, your remark is now included.)
Swatches store easily and will provide repair yarn later.
This is a matter of personal preference. Some people always re-use their swatches to save yarn; others never reuse their swatches for a variety of reasons. I fall somewhere in the middle. I usually re-use my swatch for smaller items that only use a ball or two of yarn, items like hats, mittens, and socks. For larger items (like sweaters) that require many balls of yarn, I usually save the swatch and only re-use it if I run out of yarn.
One reason to save the swatch for larger projects is that gauge is much more crucial than in a smaller project. Sometimes it's helpful to have the swatch around later for troubleshooting. It's also worth taking the time to wash and dry the swatch, whereas I don't usually bother with that for a smaller project. Knitting and washing changes the texture of the yarn, which might make it work up at a slightly different gauge unless you take the extra step of unravelling and relaxing the yarn. Also, in order to wash the swatch you have to break the yarn and separate it from the ball. Once the swatch is separated, it's easier to set it aside and start working from the ball. But if the swatch is still attached to the ball of yarn, I find it easier unravel it and reuse it.
Another reason is the amount of yarn in the project. For a small project, I usually only have one or two balls of yarn. The amount of yarn used in the gauge swatch might make the difference between having enough yarn to finish the project or not. For larger projects I try to always have at least one extra ball of yarn, so it's pretty unlikely that I'll need the swatch to finish the project.