How do I test whether all my hardware is properly working? Is there any standard tool for error checking, benchmarking, etc for checking hardware?

In Ubuntu we have such option with an application named as System testing. You can open it from unity dash by typing as System Testing.

After opening here , you can perform all your required tests. Its having everything you need.

Update : if you haven't found it already installed in your Ubuntu , you can install it with

sudo apt-get install checkbox

Hope that helps.