Connecting router to external USB drive

> 1) I don't know if I can use it to backup via Time Machine or only for
> additional storage?

   What's the difference?  Time Machine backup files are only files.

>  I followed the Time Machine guide to partition the USB drive.

   Which "the Time Machine guide" was that?  How, exactly, did you
"partition the USB drive"?

> I'd like to be able to store backups on it. Is this possible?

   The existence of "the Time Machine guide" would suggest that.  Given
the number of complaints in these forums and my own (very limited)
experience, I wouldn't trust it until I'd tested it myself.

> 2) I attached the USB drive to my router and reach the point to
> connect to server and keep getting an error that my MacBook cannot
> connect.

   Are you talking about using the Finder's Go > Connect to Server...?  
If so, then what did you do with it?  Browse?  Specify something like
"afp://readyshare" or "smb://readyshare"?  Or System Preferences > Time
Machine > select Disk...?  Or what?  Did you provide any name and
password credentials when requested?  What was the actual error message?

> I opened Genie and everything is good, [...]

   Ever helpful, Netgear uses the name "Genie" for more than one thing.
I'll assume that you're using the web interface, not some application
program.

   Define "everything".  Define "good".  As usual, showing actual
actions with their actual results (error messages, ...) can be more
helpful than vague descriptions or interpretations.