The Company Headquarter is where all the paperwork for your company is processed It's not necessary until your company reach 50 buildings, but for many reason that will become clear as you play, you may want to build it sooner.
Why can't I build wherever I want?
That is all up to the Mayor. Just like in real life, you need to obtain a license to build on a piece of land. In this case, the town's mayor decides what goes where according to his zoning rules. To exercise that option, you must be mayor.
Even when I have the headquarter I still can't build the buildings that need.
Once you have built your headquarters, it only performs basic research. Click on the Research button, you will find a list of things you can research there. The first research of a specific industry or commerce will enable you to build the relative building.
Can I build on someone else's territory?
Yes, you can. But you have to take the fact that you might be competing with that player into consideration and if they have better facilities, your building may not become profitable. If you do choose to go up against that player, you need to be on top of what the town needs and be able to build whatever that is before the other player does.
Are there any entertainment facilities?
Yes. In your Build menu, you will find shops, as well as, entertainment facilities. Some may need to be researched.
What other types of facilities can be built?
You may build residential, commercial (shops), industrial (factories), offices, special facilities (you'll find out what) and public facilities (hospitals, schools, parks, hospitals, statues, police stations, etc...)
If I decide that I don't want my headquarters anymore, can I demolish it and still get all my money back?
If you decide to demolish your building, you will only get a refund depending on your level. If you are an apprentice, you will get a full refund of the construction material you bought from the Trade Center. But if you are an entrepreneur, the refund is only 2/3. After that level there is no refund.