Home office only at home or in another household?

I have a home office and the question arises, can I use my laptop to work on what I also work with my friend moving in for a week and from there?

Probably not. Your company and your apartment will almost certainly be registered as a place of work. So you are obliged to work either at work or at home. There are also technical reasons for this

I have no employment contract in what is from Home Office?

And yet your employer decides where to work, so you have to ask