Yes, you have to be close by in order to gain experience for what your minions kill. If you're half-way across the map you won't get anything.

If you just stay back and heal with Sedna, you're doing it wrong (imho). She has exactly one heal spell. Her HP regen from her passive is rediculous, and she has pretty solid health the way most people build her. Her pounce, one of her foremost moves, is practically melee range, and the alternative would be to go for Yetis, who are considered useless pretty close to across the board. So I wouldn't recommend you stay back unless you are seriously low on HP.

As for the minions, you do get XP for their kills, provided you are close enough. This prevents you from summoning your minions and sending them off to a different lane, thus farming twice as much as the average player and quickly gaining levels that assassins have literally no ability to catch up to. If I remember correctly, you also get gold for your minions' kills, so it's even better I think the responses so far have it pretty wrapped up.

experience range is 30. its quite a large range, but still requires you to be relatively nearby. you don't have to have dealt any damage to a creep to get exp from it, just be within 30 range units when it dies and you'll get some exp.