Could the "standard online Edition" be installed on computer or it is to be used online only? Could this version be used to fill a small self-employment contract?

Yes, you most certainly can use the Standard version to enter your Self-Employment income and expenses, and this is how you do it. Once you have completed the "Profile" section of your return, you will find a menu on the left of your screen. Here you will go to the Income tab and then "Self-Employment" and start from there.  

No, you cannot install the online version on your computer; which allows me to think you are not certain of the differences between the online version and the downloaded version.  Please review the link below to assist you in understanding and making the choice that best suits your needs: