Not really. I didn't at all on my Wiz, I had only had 500DPS or so going into it, along with under 2k hp. I got royally owned by a few elite packs (mortar + molten = wtf ouch). Mortar + shielding was pretty rough too. So far I've managed to make it through everything, almost up to SK now. I only played for about an hour and a half last night, so I'm not moving much slower than expected despite no farming and a lot of pretty mediocre gear (Most of it pre-30).
You may need to swap skills around specifically for certain elite packs, but it should be doable.
No, farming for the jump from normal to nightmare is not at all necessary. I guess if you totally avoided any and all vitality gear pre-nightmare than you might have some trouble with elite packs and champs, although you'll still have some trouble if you get unfortunate combos (jailer+mortar? OH GOD NO).
Anyway, I didn't need to farm at all, and unless you geared up completely negative vitality, I wouldn't worry about it.
Act 1 nightmare is easier than act 4 normal thus far. I'm about 500 dps on my barbarian. Everything up to the SK was easy. Sometimes you'll get a pretty crap set of elites (vortex, poison or arcane, waller) which suck. But it's totally survivable.