I'm not sure about riding on the beach - there are stables in Huntington Beach but I live here and while I see horses around the neighborhood I don't think i've seen any on the beach
But I see you're also asking about what to do in Hollywood - you'd be near Griffith park up there and I've seen horses being ridden in the park, which is lovely. Lots to do there - a cool train museum (traveltown) with a bunch of old trains you can look at, plus the Griffith observatory from which you get a great shot of the Hollywood sign and great views of Hollywood - plus a bunch of films have been done up there ( Rocketeer and Rebel without a Cause - there's a bust of James Dean). 1 sunday a month Walt's barn is open for visitors too - look for the Carolwood Pacific Society online for info.
Also it's not far from the Disney Studios - you could drive around it and see the dwarves holding up the roof of the main building and the watertower with Mickey on it.