Any reasons that prevent Apple Remote Desktop to wake sleeping Macs

Apple Remote Desktop will involve either having a VPN or open ports and static IPs. I think you will be much better off with LogMeIn.com's free version, which will require zero intraction from your family members and open ports are not necessary. You will need to know a user's credentials on their computer. If they require more privacy they can disable LogMeIn from the menu bar item, as it will always be on as long as they are connected to the Internet. I suggest adding a user to each computer with your name and one password—that will be one less thing for you to remember and if they change their password or don't remember it you can still help them.

I use both ARD and LogMeIn daily, but ARD is more than you need.

By default, Mac OS uses VNC combined with a few extras (e.g. auto-discovery via Bonjour) when you use Screen Sharing, and in most cases this is all an average user needs. For system administrators however, there is usually a need for more control over the computers they support; therefore Apple provides "Apple Remote Desktop" (ARD) which offers a lot more control mechanisms (like terminal sessions, remote installation of software, remote shutdown, remote execution of automator scripts) and this is possible with groups of many computers at the same time. For this feature to work, SSH and remote management needs to be activated.

So in short: Screen Sharing is roughly equivalent to VNC, while Apple Remote Desktop is a "high-end" remote administration software which requires "Remote Management" to be turned on. There is also a difference in how incoming screen sharing requests are handled: with "Screen Sharing" enabled, if an observer connects with a user account that's different than the one currently logged in and clicks Share Display, the user needs to confirm before the screen is shared. With "Remote Management", the screen is immediately shared without a prompt appearing.