Yes, placing second district in the Palace tile creates a quarter that gets all benefits from wonders, policies etc.
When it comes to second question - there is no simple answer i think...
it's worth to place the buildings you put in your example where they get most benefit from adjacencies... but you should still at least try to cluster your quarters. Better adjacencies means better specialist etc. You also have plenty of possible quarters at your disposal. We have 13 building in antiquity and mordern age and 15 buildings in exploration era (without counting bridges, unique districts and warehouses!) so you can still build around your palace, even though you have to reach 1 or 2 more distant places.
The rule I try to follow is to not place the wonders next to palace, cause it won't get adjacency bonus from it, unlike ALL the other districts. I try to put the wonder in the second ring of my city and build around it and city center.
Here is an example so you can understand how I start to plan my cities.
(Sorry for using Paint but without in-game pins, that's the best solution for me right now
)
So, in this example first thing i do is i take a look at my surroundings and look for good spots with great adjacencies for my quarters. I see I have 2 spots for my Science/Production buildings, 2 spots for culture/hapiness and 1 spot for gold/food. I put them on the map aswell as my palace (white colour).
That leaves me with 3 spots around the city center and one in the second ring between the two quarters. If I plan to build a wonder, I want to maximize the adjacencies that it will give me. When you look at the free tiles it's easy to decide where to place it. Next to the Elephant resource, i have a spot that can be surrounded with 5 quarters (or 10 districts). Let's mark this spot for a wonder placement.
I look now what buildings are left and according to that information I plan the rest. If my civ has an unique district, that makes first choice very simple. I will put it on a tile between wonder and city center(black colour). I also still have one coastal tile left next to my wonder and I have Lighthouse at my disposal so let's put it there. What's left are 2 spots around my palace - i can put there a warehouse building if there is an obvious choice for one (not like in this example), Altar, or I can even wait for next age to get 2 extra gold buildings (2 yellow markings on the left side of the city centre) for which i dont have a spot with good adjacencies for.
This way i have filled 5 out 5 spots around my wonder, and (without altar and warehouse) 4 out of 5 spots around my city centre while still putting districts in the best possible place.
Tbf, to answer your question shortly - you should hunt spots with good adjacencies AND build around palace at the same time. Of course it's not always possible, but you still have to try to go for it.
I hope you get what i mean
sorry for such a long answer. and I still haven't talked anything about civ specific bonuses that can change everything (like Han bonus to adjacency for quarters on your Science buildings)...
hopefully someone will be able to explain it in shorter fashion.
In my most scenarios in order to grab those adjacencies I need to grow my city in a line! and have like 3-4 quarters from one side of palace in a line and some on other side etc... and I wonder if its worth doing this instead of just build on first ring next to palace. For example a science building next to 2 resources gets +2 science, while next to palace gets +2 yields too (1 science and 1 culture). So I guess its worth settling away from palace only if can take 3+ adjacent yields.
