1. By "filesystem sizes," I'm guessing they want to know the sizes of your partitions. In Windows, this can be done by going into My Computer, right-clicking each local disk, and then clicking Properties. This will give the total size of the disk or partition (Windows does not distinguish between the two in a user-accessible fashion). I'm guessing your support team wants to know the sizes of each of these.
2. The directory in which most applications will be installed in Linux is /usr/bin. On big multi-user systems, this can also be /usr/local/bin. The difference between the two is that /usr/bin contains most of the general-purpose and system-wide apps, whereas /usr/local/bin contains more specialty apps. It really doesn't matter which, as long as users can access the app in question.
Hope this helps