While you do have to pick up the pace in the Demon Halls due to the spawn rate (especially floor 24, dear god) you can't just skip them. You need to explore for... reasons. And not just for the 3 upgrade scrolls and 2 potions of strength.

As for Evil Eyes

• You get 2 turns once they start charging, sometimes more if they are chilled. Only fight them when you are near cover and can duck around a wall.
• They have enormous damage reduction while charging. Don't bother trying to hit them when they're charging unless you only need one turn to reach cover, in which case a couple damage is a couple damage, I guess. Setting them on fire or using a Wand of Corrosion on them before you seek cover is good.
• Their laser destroys doors, but the door provides cover.
• If you can blind then on the first turn they charge up and then step perpendicular to their line of site, they'll shoot and miss without you needing cover.
• If you freeze or paralyze them while charging, they will still blast you when they can move again.
• It does suck to get blasted from the shadows. Buy all the torches you can.

Make sure you don't let a ripper demon jump right on you. The bleeding causes so much damage! Other than that, they are pretty squishy. Even if they are in the shadows, the game warns you they are about to leap and flashes a little target on the floor for a second. Try to step away from the target. They always target their jump in the direction you just stepped (assuming you weren't standing still). Sometimes you can cheese them into jumping into a pit, but it's really tricky.

The succubus will teleport to you and get a free attack (it has a cooldown, though), but if there is only a single tile between you, she will just step toward you. So if you're kiting her with a Ring of Haste or she's chilled, don't get too far away. And if you have primarily ranged attacks and she's sleeping, you're often be better off walking up to her until you are only one tile away from her before attacking with your ranged weapon. You only get one unanswered attack in that case (if you don't have haste), but that's actually better than being attacked the instant you wake her. She also has a max range, so if you can attack from very far away, that's better than everything I said, but that always seems rare.

Save Scrolls of Magic Mapping for the demon halls. They are super good there.

I'm guessing you're talking about the evil eye things that shoot lasers. I HATE those things. The good news is that they take 2-3 turns to charge up, and the laser doesn't pass through doors or walls so you have time to get away.

Getting that far is really good for just starting

This is the furthest I've gotten as mage. Had good on lock with horn of plenty and warlock kit. Thought I finally had it and got wiped seconds after getting to that level.