While I am not a native English speaker my understanding is that the tense of present progressive (He is writing a book) is used when someone is doing something right now/at the moment. Thus, it is not necessary to add the word "now" on top. Instead, thevword now used in Czech should be the indicator to use present progr. In English...
I (non-native speaker) understand the present continuous as a description of something happening in the current time. Saying "I am writing a book" can mean for me both that I am sitting at the desk with a pen in my hand as well as something like "I started some time ago, the process is ongoing, end is expected in another month - but at this very moment I am enjoying a BBQ with my friends. The word "now" clarifies to me that the first option is the correct one.
I'm a native English speaker, and I've seen a few of these present simple / present continuous debates— it's my opinion that both should absolutely be accepted, in all cases. Sure, the present simple may be less common in many cases, but it's absolutely correct (and in some instances, necessary, especially when the activity is being discussed as a present activity but isn't actually happening). For instance, if I were proposing a schedule for the day with a co-author, I might say "Why don't we write the book now, and get lunch after." Or, if I was a daycare instructor and naptime was beginning, I might suggest to a coworker that "Zofie sleeps now" even if she is not presently sleeping.