Does Lightroom only work when I am online? Do I have to be online to process or work on files? If I cannot be online, eg, when in a remote area with no service, how do I process my photo's?
No, Lightroom works on your local computer. If you have subscribed the creative cloud then the Internet has to be accessed on a monthly basis to verify that you paid your fee. But other than that everything is totally local.
You can easily test this by cutting off all network access to your computer (turn off wi-fi or disconnect the network cable). You'll find that you can still get stuff done if there is no Internet. Especially if you are using the non-subscription version of Lightroom, which doesn't need to check in at all after it's first licensed.
If you do use Lightroom through a subscription to Creative Cloud, the annual plan has an offline mode limit of 99 days (over 3 months); the limit is 30 days offline for the monthly plan. And if you stop your subscription, you still have access to your images and some Lightroom features.