Sicily - one week itinerary advice?

Yes, you can drive to all of those places. The question is whether you'll have time to see much when you get there, and whether the time allocation is optimal for your interests. You haven't told us what those interests are.

You're changing hotels at least 5 times in seven days, which I'd estimate to cost you about 2-1/2 days' worth of your very short vacation time. If you're flying in from the US or Canada (unclear from your post), your first day in Palermo may find at least some of you brain-dead and incapable of doing much beyond trying to stay awake, so I don't know whether you have 6 really usable days or closer to 7 (depending on what time you get to your Palermo hotel on Day 1).

For me, 2-1/2 days for Catania/Etna/Taormina vs. about 1/2 day for Siracusa is seriously out of balance. But that's because I really like Siracusa and the baroque towns near it and think 1) Catania (while worth a visit on a longer trip) sort of pales beside Siracusa, 2) Taormina is gorgeous but swamped with tourists and sets my teeth on edge after a short visit, and 3) Etna can be seen from a distance while you're in the area. Opinions differ, obviously, and there's no dispute that Taormina is beautiful.

I haven't been to Agrigento myself, so I have no idea how much time is needed for a reasonable visit there.

Part of the problem is that you seem to want to visit mostly the eastern part of the island but have opted to fly into Palermo rather than the larger Catania airport.

I agree, your plan is very aggressive.

Can you change your ticket to fly out of Catania? You would not have to backtrack to Palermo. And if you don't get close enough to Mount Etna, you can get a great view of it from the CAT airport boarding area.

Do you plan to change hotels for Taormina-Catania-Siracusa-Mount Etna? They are not far apart, you could base in Taormina or Siracusa (which really should have two nights).

I had a very aggressive trip to Sicily, and afterward I regretted it. I jam packed each day to the fullest and the only time to wind down was at dinner. Unfortunately, I didn't sleep well at night either so that only added to the fatigue. If I were you (and it sounds like flying into and out of Palermo is inevitable), I would focus on the western part of the island only instead of trying to see both sides. Flying in and out of Catania would have made more sense for your current itinerary. If you pursue this, at least use Catania as the base for Mt. Etna and Taormina since each are not too far and it will save you from having to change more hotel rooms.

Etna is iffy - even when the rest of the island is sunny, it can be enveloped in fog and cloud. Or it could erupt, like it did a few days ago. I didn't get to Taormina but I've heard it's lovely but kinda touristy. I loved Syracuse, but you've only got a week. So my recommendation is to stick to the western side of the island. There is plenty to see, the temples of Selinunte and Segesta, Monreale, Cefalu and the coastal towns like Erice and Trapani, maybe even venture as far as lovely Piazza Armerina and the wonderful mosaics at the Villa Romana de Casale. Spend 2-3 nights in Palermo, see the city's highlights, Monreale, maybe a day trip or overnight in Cefalu by train, then rent a car and stay 2-3 nights in a coastal town or an agriturismo near Selinunte and day trip around the area. Then Agrigento for a day, overnight in Piazza Armerina, visit the Villa, then drive back to Palermo.