Is there public parking close to Grauman's and Walk of Fame? Can we drive and park ourselves instead of having to take a tour?
Yes its very easy! Personally for the price I suggest the Hollywood and Highland center parking. You get off the 101 at Highland, follow down and on your right, immediately before Hollywood Blvd. you will see a parking garage (just past the Renaissance Hotel), I believe its 2 hours free and they validate your parking at the shops and restaurants in the complex (pick the place with the HIGHEST time validation since some places give you more then others).
The complex is home to the Kodak theater, then if you walk through the complex as I mentioned above to the street , across the way you will see El Capitan/
Disney Store/Malt Shop, if you make a right and head up the street a little ways you will see the Chinese Theater and while you walk you will be on the Walk of Fame.
For those who haven't been their and have kids, watch out for the dressed characters. Some of them look really cool, but none take a picture for FREE!! They all expect money. So if you don't want to pay don't try and take posed pictures.
PS- If I sound partial to the El Capitan I am!!!! Disney does an AMAZING job of showing new or special Disney films there and they always add an extra. They have had full sets out to see (Haunted Manison and more), props (NBC and others), full dance numbers (several including Enchanted), Princess Tea Parties (Princess Diaries). These happen all the time just check the website. The theater is also stunning and has a live Organ player, that alone is worth the price of admission (pretty much a standard movie ticket).
http://disney.go.com/disneypictures/el_capitan/
ETA: Don't park on the other side of Hollywood, or up Hollywood in the standard lots they charge $10-$15, and don't validate.