If I'm not at home (with my console) can I extend the family timer remotely? On the xbox360 can I log in away from home
Yes, you can do it via xbox.com
You can change the online safety and privacy settings for any profile on your console. Here’s how:
Change your safety and privacy settings
Go to the My Account page. At the upper right-hand corner of the page, click Sign In to sign in to your Xbox Live account. Under Settings and Preferences, click Privacy and Online Settings. Change the settings to whatever you’d like. picture Change your child’s safety and privacy settings
Go to the My Account page. In the upper-right corner of the page, click Sign In, and sign in with the Microsoft account of the child whose settings you want to change. Under Settings and Privacy, click Child. Note You will be prompted for the parental Microsoft account in order to make changes to the child account’s safety settings.
Change the settings to whatever you’d like. Disable Internet browsing on Xbox for a child account
For information about how to disable access to Internet Explorer on your Xbox 360 console, see the Restrict your child’s access to Internet Explorer on Xbox Live page.
I've looked into it a bit, and I can't find evidence that a remote session to your xbox is possible. The only thing you can do without touching the console is add games to your download queue, which will then start downloading when you turn it on (physically).
If you use a child account on the xbox you might be able to edit some settings on xbox.com. I can't check that for you, because I don't have a child account.