Go read Sheldon Brown's site re: fixed gear.
1. Yes you can ride fixed off road, but it will only be fun if you already know how to ride fixed.
2. Fixed off road isn't as fun as single speed.
3. Yes, use foot retention (and brakes) with fixed gear on road. Probably doesn't matter as much off road.
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Be watchful, stand firm in the faith, act like men, be strong. Let all that you do be done in love.