Starting a software company without registration

I want to sell a software product. I would like to use a brand name e.g. mysoftcompany in the EULA (End User License Agreement) and in the contact details of webpage. So that the software is sold under a name of a company.

Is it necessary to register such a company before using it since I have no office or physical establishment? If this is not possible, can I use the name of my website e.g. mycompany.com instead of my name in the license?

Registering a business and have an office are two unrelated issues. If you want to use a name, then you can either use your own (so you are personally assuming ownership of any and all agreements and liabilities resulting from your activities), or you can create a business (in which case the business would assume the same, pursuant to the nature of the business structure you create).

If you are going to sell anything and have any kind of EULA, you should probably form a business entity to own that work. This has benefits both from a liability perspective (what if your software has unintended consequences to a user and you get sued) and from a revenue perspective (tax planning).

State law provides the rules on use of business names and assumed names and they vary from state to state. To use a business name, which indicates a business entity, you most likely will need to register the business. You can probably do business as yourself (or as a business entity) under an assumed name, but you will need to file a DBA (Doing Business As).

Using a name other than your personal name without registering it as a business or filing a DBA is probably against the law in your state.

Note that registering a company or filing a DBA means your state will permit you to do business under that name. Your state permitting you to do business under a name does not mean that you have a right to use that name under federal copyright and trademark law. That is a separate issue.