How the hell am I supposed to survive in the underworld?

Find a place where below you there's a lot of ground (ash) and hellstone (no buildings or anything where the imps can spawn on). Dig a tunnel (width 2) under you (block fireballs with a fast pickaxe) and mine until there are no visible places to spawn for imps, other than your own tunnel (which you can even fill with dirt).

Then it's easy to mine hellstone while blocking the fireballs of the few imps with your pickaxe (always face the enemy while mining). There won't be many imps since they don't have much space to spawn.

A stack of potions, magic mirror and any flail (for worms) should then be enough.

(I think the tunnel in the image is too wide, but hopefully this still makes it more clear)

Patch 1.0.4 increased the spawn rates and maximum spawn numbers quite a bit, and the Underworld wasn't a cake walk before that.

Some general tips for surviving:

• Take less damage to avoid dying.
• Bring some excess dirt and cover up all the lava in the area, first thing you do.
• Bring a lot of health potions. At least one, maybe two full stacks.
• Bring a powerful flail-type weapon, such as the Blue Moon, to deal with the Bone Serpents. If you've already got a Sunfury, even better!
• The Muramasa is also effective against Bone Serpents, if less so; but, it helps tremendously with the Fire Imps: Its high swing speed will allow you to avoid most of their projectiles while charging at them, especially if coupled with Feral Claws.
• Put off dealing with the Fire Imps until they overwhelm you or a Bone Serpent spawns. If you swing your Nightmare Pickaxe constantly, you should be able to swat most of their fireballs out of the air while you're mining the precious Hellstone. For this purpose, the Molten Pickaxe is not recommended, since it swings slower.
• If you've established an escape route, you could try building a minimal shelter with a Bed, so you respawn right where you were slain. If you happen to run out of health potions and cannot escape, simply tear down the Bed to revert to your original spawn point, then return by Magic Mirror or dying.