How should I log total hours flown in different countries?
Is it mandatory to bring your flying hours from the European logbook to the "AMOUNT FORWARD" section in the American logbook?

No, it is not mandatory to bring your flying hours from the European logbook to the "AMOUNT FORWARD" section in the American logbook.  

In the USA, logging time is not required at all other than information for required recent experience and check ride prerequisites.  Also, there is no reason the EASA logbook is not an acceptable document for that purpose so long as it contains the needed information such as landings.

What is a good practice if you have multiple foreign logbooks?

Consolidate them into one logbook that best fits the system in which you do most of your flying and keep only specific additional requirements for the other systems logged elsewhere.  

If you need to keep the total flight hours separate, I would use one of the customizable columns that I have seen in most logbooks and label it with whichever system you fly in less.  Then you can simply subtract that column from your grand total to deduce the other system's total time. 

Although, based on how much I get picked on by my German friends for not using the metric system, I could understand the assumption that we must be different.