We didn't play it like we usually do--I encouraged my players to just do whatever they wanted. Since I had no plot or setting to ruin, I figured it wouldn't hurt. It was insane. They got in a bar fight right away.
You've got good players. There are two typical reactions to sandbox play: (1) insanity, and (2) sit and stare at the DM in silence. #1 is way more fun.
Suffice to say everyone died.
Good session, men. Good session. <
nods>
We actually played our normal 4e campaign later in the day, and I have to admit, I enjoyed the RC game more.
Hook, line & sinker ...
And yet you are well spoken, and have clearly superior taste in RPGs. I expect great things of you.
though our little on-the-fly game lacked a lot of what our 4e game has (mapped out areas and regions, predefined NPCs, dungeons and adventures), it was more immersive.
And good books are more immersive than movies. The imagination has more colors, and fine detail, than any pre-painted mini.
1. Is Keep on the Borderlands a good way to start?
Yes, but too late; you already started. And most excellently!
I ordered it off Amazon the other day. My hope is to start a monthly sandbox style game. Was this a good decision, or would something else have been better?
It's certainly a good decision, but
don't lose that freedom from the first session! It's a springboard, not a straight-jacket, and you should ignore any part of it that gets in the way of having a blast.
2. Do the PCs have no hope for life? Though one of my player's mottoes is "I LOVE DEATH," I'm worried the brutality of the game may turn off some of my players. One player in particular doesn't like to "lose." I'm worried that it seems a single goblin can kill a level one character in one hit. I'm worried they'll just run from everything, get annoyed at having to roll new characters all the time or, the worst, get downright angry and frustrated over their characters dying. Are there any ways to help with this? Or does it matter?
It's a different style of game. It's supposed to be that way. Perhaps apocryphally, EGG recommended not naming your PC until 4th level. It doesn't help them survive, but it helps you move on!

They'll get used to it, but you may have to help them make necessary mental adjustments.
It is becomes a real problem though you may wish to introduce an HP kicker like some other posters, Negative HP Death & Dying rules, or something more exotic, like an
Injury Table.
but RC is just cool in a way no other game I've played has. Maybe I just haven't played enough to discover its flaws, but it was some of the most fun gaming I've ever experienced. And it was only one short session.
That's why even though Monte Cook and Mike Mearls are smart guys, EGG was a genius. He got it right.