SQL Server 2008 R2 Express size
The "Database Size" on the properties page is showing the size of the data file.  This will be the size of your data, with some empty space for new data.  You can configure the rate at which this empty space is allocated in the database properties.

It says space available 0.03Mb because you've shrunk the data file down - it will grow by itself to allow for new records - this is normal and intended behaviour.  This "Available Space" is what's left in the current data file, NOT the difference between your current database size and the licensed limit of 10GB.

You can verify the edition that is installed by checking the General page in the Server properties under Product.  In the management studio header it only shows the general version, not the product edition.

When you delete a lot of records, you will end up with free space left in the file.  This will not be automatically shrunk unless you set up a maintenance task for it.