I guessing that the cemetery shown at the end of the movie is the Inglewood Park Cemetery. It is located across the street from the L. A. Lakers former home, the Forum. The cemetery is huge and has changes in elevation so that cameras can look down from a rise without showing buildings outside the cemetery boundaries. I'm familiar with the place, having attended my grandmother's funeral there.

The LA scenes:
The pick up Flynn at Perinos on Wilshire Blvd.(demolished 2005)
They do come down the steps of LA City Hall in one scene.
They drive around Los Feliz en route to Allison's apartment (2040 Rodney Dr.- it hasn't fared well)
Leaving Allisons they go over Glendale-Hyperion Bridge
Flashbacks to the party with Allison while theyre driving- Max walks through the bathroom of the Oviatt Penthouse and sees Allison in the bedroom there (the furniture seen there is original to the apartment- still there last time I saw it).
Then they have hot dogs at 4th & Main (across the street from the Barclay Hotel and Farmers & Merchants Bank)
Max's house is also in Los Feliz (2073 N. Catalina St- looks more or less the same)
Mulholland Drive scenic overlook is another location, seen by day in the outdoor scene with Allison and Max.

several miles north east of downtown L.A.

Here is a forum I frequent
http://forum.skyscraperpage.com/showthread.php?t=170279
They've been compiling pictures of LA for the last few years, it's one of those forums you can spend hours scouring. I'll bet some of them could tell you where a lot of it was filmed.

When the four of them are driving in the car to the motel where Jennifer Connelly lived looked like Los Feliz/Silverlake area.

I went to Hollywood not too long ago and took the Dearly Departed Tour. One of the houses went went by is Nick Nolte's house in the movie, the one he walks out of after Melanie Griffith finishes watching "the film." Chazz Palminteri is there waiting on him and jumps in the back seat of Nolte's car saying he isn't going back to the captain and saying he can't find Nolte. I believe that was Los Feliz and the house is not too far from the Jaws House where the Black Dahlia supposedly died at the hands of Dr. Hodel. (That house was also used in 'The Aviator' as Ava Gardner's house.)

I think a lot of the residential scenes were Nolte and Melanie Griffith lived were shot in Los Feliz, a section of LA betwen Hollywod and downtown. Lots of Mediterranean and Spanish style houses from the 1920's and 1930's there.

Well, without looking at the location page, I can tell you that they used City Hall. There's a scene where Nolte walks out, meets with the rest of the squad, and in a long tracking shot they cross the street and get into their car.
Edit: Okay, I just went and looked at the locations page. They have city hall listed, and a couple of city names. Wow, really useful feature, that is.
Unfortunately, I didn't recognize any of the other locations.