Is there any problem with downloading and overwriting modules?
You don't need to disable a module. That is often not even possible e.g. when there are many other modules relying on it and disabling them would break your site.

You might want to take your site to maintenance mode while running updates though, as your site might completely break between replacing the files and running update.php. 

What however is recommended is to remove the old files first and then add the new directory. The reason for this is that you otherwise end up with stale files if files have been moved or deleted.

If you have shell access, I can strongly recommend using drush dl/up for this task as this will handle this for you automatically.