Vivekbrata Basu:
Welcome to the Forum
I'm confused by your itinerary, but most importantly, I'm wondering why you've made Hollywood the center of your visit. It's true that this district (which is one of OVER 100 in the city of Los Angeles) is well-known but focusing on Hollywood is like visiting London and focusing on Leicester Square (if only Hollywood was as nice as Leicester Square).
Hollywood is an older part of Los Angeles (okay: 'old' is kind of vague when one compares the age of, say, London) and the district fell into disrepair in the 60s and 70s. There was an urban renewal of a sorts in the 90s and it does look a lot better than it did but it's not a glamorous place; there are no celebs roaming around in their Rolls Royces here. This is a low/middle class neighborhood with drab 1930s-era office buildings (serving the burgeoning movie industry of 100 years ago) with an occasional historic theatre thrown in. The Hollywood & Highland Center mall is the 'big change' in the neighborhood and, for what it is, it's appealing especially with its motif that salutes the Golden Age of cinema.
But overall, the neighborhood is 'nothing special' (I wouldn't call it seedy but it's certainly older and unremarkable). I, too, had to look up what the La Brea Gateway is and learned that it's the 4 statues of ladies on the island in the roadway where Hollywood Bl & La Brea Av intersect. Perhaps, these ladies are something special. I confess that I've walked by them (and even looked at the signage relating to them and do not recall anything about it).
The rest of the attractions in this area are 'touristy places' (the wax museum, Guinness Records Museum, Ripley's Believe it or Not!), etc. These are not especially interesting to me; you might feel differently. In general, a glance at the courtyard of the Chinese Theatre and a brief walk along the Walk of Fame is probably all you really need to get a taste of Hollywood. That's a visit of about 60 or 90 minutes. If you want something more formal, take a 90-minute walking tour of the area with RedLineTours.com
Apart from that, I would encourage you to explore the second-largest city in America. Thus far, the only place in the Los Angeles on your radar is Hollywood (Beverly HIlls and Burbank are not even within the City of L.A.). That's not to say the WB is a bad place to visit (it's not) but there is SO MUCH MORE to Los Angeles than Hollywood.
Here is a list of several hundred attractions in and around Los Angeles. Check these out and see what might appeal to you.
https://www.tripadvisor.com/Attractions-g32655-Activities-a_allAttractions.true-Los_Angeles_California.html
Link and SCROLL DOWN to the numbered list of attractions.
You'll be heading north to L.A. from San Diego so maybe you'd enjoy stopping in the San Pedro District (L.A.'s port) to check out:
a. The Korean Bell of Friendship
b. Various whale watching Tours
c. The Battleship USS Iowa.
Or maybe stop in neighboring Long Beach and visit the RMS Queen Elizabeth or try out some standup paddle boarding or Kayaking in and around Naples Island -- which is packed with luxury housing positioned along a circular inland waterway. You can paddle through this canal or walk along its banks.
All of these attractions and hundreds more are in the above link. Do not miss out; there is no need to drive for hours out of Los Angeles for attractions. There is plenty to see and do here.
As far as your current itinerary is concerned, I'll reproduce it here and add my remarks (beginning and ending with elipses (...)
Mon 22-Nov:
Arrive LA by noon. Spend some time in the Hollywood Blvd area. Cover La Brea Gateway to Chinese Theater. ... As stated above, about 90 minutes is all you really need in this district...
1:30 PM - Drive/taxi to Warner Bros studio tour. Booked a tour at 3:30 PM. Late lunch at Central Perk cafe before tour. Tour duration approx 3 hours as per their website. ... This is where I start to get lost. First, if you have a car and are willing to drive to Joshua Tree and Ventura (for the Channel Islands) why wouldn't you drive 30 minutes to Burbank for WB?
Second, if you leave Hollywood at 1:30 and arrive at WB at 2 pm, you'll have 90 minutes for lunch (actually about 70 minutes since they want you at the Tour Center about 20 minutes before your tour). If it were me, I'd have a lunch that was memorable. I wasn't aware that they were serving lunch at the Studio but I can't imagine that the food would be anything special. There are lots of places within a few minutes of the Studio that have good food. If you like (or want to try) Mexican, consider Don Cuco (the location on Riverside Drive, a short walk from the Studios), for example.
Alternatively, instead of touring Hollywood until 1:30 pm, leave Hollywood around 1 pm (since you can tour the district anytime -- the Walk of Fame and Courtyard of the Chinese Theatre never close) head to the Beachwood Canyon neighborhood (10 minutes east of Hollywood) and stop at Birds (a unique chicken restaurant) or one of the other restaurants in this district) and then head to Lake Hollywood Park and have a look at the Hollywood sign close-up as well as a view of Lake Hollywood (across the road from the sign) and then head toward Burbank.
Here is a map of such a drive: https://goo.gl/maps/PFC3KeR99xqZWS9E8 ...
7 PM - Arrive back at the Hollywood Blvd area. Spend time between Chinese Theater and Egyptian Theater. Have dinner. ... Or, head back to Los Angeles via the Los Feliz neighborhood which is a much nicer place to explore on foot. There are lots of dining options along Vermont Av and Hillhurst Drive. (two of my faves are Palermo (Italian) on Vermont and the Alcove Bakery on Hillhurst (Southern California fare plus awesome cakes -- try the lemon coconut or the princess cake (marzipan and raspberry -- yum). The drive back to Hollywood is only about 15 minutes. Here's the route from WB back to the hotel via Los Feliz:
https://goo.gl/maps/1LYnNBDxfHuvenFa8 ...
Stay in Hollywood Celebrity Hotel (booked). ... This is a BASIC 2-star property with very good ratings. It's not fancy but it's well-managed.
Tue 23-Nov:
8 AM - Leave for Griffith Observatory. Spend 1.5-2 hours. Our primary objective is to see the Hollywood Sign from there. Leave by 10. ... If all you want to do is see the sign, you've already seen a better view of it from Lake Hollywood. Pick something else from the link above.
In fact, you're apt to find lots of things to see and do in the link above and when you do, you'll give up one of your excursion days to spend more time in L.A.
4 PM-ish: Head back to LA. Spend some time in the mall or tour the Hollywood Wax Museum and the Guinness Museum. Or just take it slow in prep for next day AM. .. See my comments above...