Naples, Sorrento, Sicily

I'm planning to spend (alone) about two weeks in the Naples, Sorrento,
Amalfi Coast area and another two weeks in Sicily. I have some
questions:

1 - I'd like to swim at the beaches in Sicily. Will the water be warm
enough in May, or should I wait until September?

2 - All the travel guides (and posters here) recommend staying in
Sorrento and just taking a day trip to Naples. But it looks like the
sights I want to see in Naples will take at least three days. I
realize Naples is crowded, bustling, and (allegedly) crime-ridden, but
is it REALLY such a bad idea to stay there? I could stay in Naples,
e.g., 5 days, seeing Naples and Cumae, Pompeii, Herculaneum, Vesuvius
-- then move on to Sorrento to stay for several more days, seeing
Sorrento, Capri and other islands, Positano and the Amalfi Coast, and
Paestum. It just seems that would be better than trying to do
EVERYTHING out of Sorrento, spending a lot of time commuting to
Naples, missing Naples nightlife, etc.....

3- I had planned to take the train from Sorrento to Sicily, but now
I'm reading that that is not such a good idea -- it's a long trip, the
trains are uncomfortable and schedules undependable (unlike those I've
ridden all over northern Italy). I've been told that a boat or
hydrofoil out of Naples or Sorrento to Sicily would be better. Is
this true? What are the relative costs and length of time? Would I
miss worthwhile southern Italy sights by doing this (e.g., Salerno)?

4 - I had planned to rent a car in Sicily, since I want to see ALL of
the island. But I would prefer to take a train between coastal cities
and bus to inland towns rather than bother with a car. Is that
feasible?

5 - Tips, things to see or avoid, recommended restaurants and hotels
(80 euro range), are very welcome. (I have Rick Steves's Italy and
the new Eyewitness Guides to Naples/Amalfi Coast and Sicily.)

Thanks to the group for your input.....