Does the operating system matter when syncing? I'm going to buy a mobile phone with android and I have use linux as operating system on my desktop computers and laptop, as android is based on linux I could hope the compatibility with linux should be okay.

I've been distant from the mobile phone industry for some time, but I've seen that some phones use their own software, does android use an platform-dependant application to communicate with the phone?
Android doesn't natively use any software to sync or perform syncs with your computer.  It syncs with your Google Account services in the cloud.  You can sync certain things with your computer in different ways and it all depends on what and how you are trying to sync, but generally, Linux machines work just fine with Android.

If you want more detailed answers try being more specific about what you want to do (sync your contacts with Thunderbird, or sync you music...).