Why did Nedry shut off the fences?
The novel covers your first point. In order to gain unfettered access to the cryo-room so that he could steal the embryos, Nedry needed to shut down all of the security.
“That's the least of it,” Arnold said. “When you turn off the
security, you turn off all the peripheral fences as well.” “The
fences?” Muldoon said. “The electrical fences,” Arnold said. “They're
off, all over the island.”
Note that he could have gotten off the island whenever he wanted (he wasn't a prisoner), but he wanted to do it in a way that offer plausible deniability that it wasn't him that stole the DNA samples. They might suspect him of taking the samples but they'd have no outright proof
Which way should he have gone?
We know that his destination was the East Docks (in the film, to meet the ferry and in the book to meet Dodgson's boat) and we know that he took a wrong turn and ended up being killed by a Dilophosaurus. On the strength of that, I've used the original 'screen-used' tour map prop to show his likely proposed route and likely accidental route.
For the record, in the movie the arrow to the docks originally pointed upwards shot immediately before but was then pointing to the left in the very next shot, before his car hit the signpost. Under the circumstance, I think we can peg this down to a production error.
Why did he have to go through the park?
Because the visitor centre is on the West side of the island and the docks are on the East side of the island. In order to travel between these, you need to cross the park however under normal circumstances you'd only travel alongside enclosures containing relatively benign dinos such as the Stegosaurus, Gallimus and Brachiosaurus.