Which types of fish in sushi must be cooked?
All of the following are generally cooked items:


Shrimp, ebi, available raw in sashimi, but almost always lightly poached.  The poaching brings out sweet and subtle flavors otherwise masked by a metallic tang.  It also prevents the highly perishable shrimp from losing texture to spoilage.
Octopus, tako, available raw in sashimi when sliced very thinly, but otherwise poached lightly. The poaching reduces toughness and brings out additional flavors in an otherwise bland meat. 
Eel, unagi, as you noted, is ALWAYS grilled, and steamed. This is necessary to achieve the truly transcendent flavor and texture of unagi.  In fact, the cooking of eel is such a challenge that it is generally a whole separate chef profession in Japan.  Most eel comes from one of the professionals, because sushi cooks know they will never be able to match the quality themselves.  Improperly cooked eel becomes tough and has unpleasant flavors.  Raw, the blood of eels is even toxic!
Sweet egg omelet, tamago, is by definition cooked.  This is not to say that raw egg doesn't have a place in Japanese food though; tamago kake gohan is a dish of raw egg on rice, with soy sauce. 
Cockle is lightly poached
Crab, kani is always cooked because... well eating raw crab meat is just vile.


For further reading, I highly suggest the online sushi encyclopedia. 



Food Safety:

Sushi is only prepared from the very freshest and highest quality seafood, and sushi chefs are trained to identify parasites and signs of spoilage.  In addition, some areas (including the EU) require sushi to be frozen for 24 hours to kill parasites.  

If that wasn't enough to help settle  your stomach, the pickled ginger and wasabi served along with sushi are more than garnish or simple condiments!   Both ginger and wasabi have potent and documented anti-microbial and anti-parasite properties.  Ginger is also well-known for combatting nausea.  Source: the CRC Handbook of Medicinal Spices, page 310 for wasabi and 316 for ginger. (links are to Google Books copy).