Can you use a screen name or must you use your real name?
Yes, you can use a screen name.
Can I keep my profile private? Or just allow certain people, like Mr. PADC and my sister for example? (I know this would defeat the purpose of the social media aspect).
Yes, you can make your profile private. You can choose who sees it on an individual basis. You can also block people from seeing your running route, or just a section of the route (like your home).
Can I choose when I want to sync my Garmin to Strava, or is it an all or nothing setting?
I use the auto-sync feature which does everything from Garmin to Strava. I think you can pick and choose, but I think the benefit is doing everything.
It sounds like Strava can be used to track runs without the use of Garmin. Is this correct?
Yes, but if you've got a Garmin, then the Garmin is superior to the phone based Strava app.
Other things I like about Strava -
-The interface is much more user friendly for the social aspect and seeing other's runs.
-The GAP (grade adjusted pace) feature is nice to see how your pace is accounted for on a hilly run vs flat run.
-Stravistix, which is a Google Chrome plug-in is a high end feature that tracks training load using HR data. It can be very beneficial, but only if all runs are uploaded to Strava.
-Strava's "Fly-by" feature which shows you and other racers on the course at the same time in a replay. So Disney races are fun because you can see where all your friends were on the course vs your location. It's neat to see how you cross paths on hairpins after that fact and whether you did or didn't see them.
-Strava tracks way more PRs than Garmin does.
-Strava has a lot of "segments" which people self-created around running routes and races that also act like "PRs".
That's just to name a few.