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今年夏天我在北京学习中文。（no localizer, just 在北京）
我刚到美国的时候只能在餐馆打零工。（no localizer, just 在餐馆）
他不在餐馆打工了。（no localizer, just 在餐馆）
BUT
我要在CSE上贴我的问题。（在Chinese Stack Exchange上）
他写了一个便条在门上。（在门上）
商店里的服务员对顾客都非常客气。（在商店里）
When is one allowed to omit localizers and, alternatively, when are they necessary?
Eventually, it all boils down to how precise the location needs to be.
Case 1: Precision is not required
這座城市發展迅速。Regarding this city, it is developing really quickly.
*這座城市裡發展迅速。
I don't really need to localise where in the city is developing really quickly. It just is as a whole. It's a generic description requiring no localisers.
Case 2: Precision is required
婚姻是被圍困的城堡，城外的人想衝進去，城裡的人想逃出來。（改自《圍城》）
*婚姻是被圍困的城堡，城的人想衝進去，城的人想逃出來。
Marriage is like a fortress besieged: those who are outside want to get in, and those who are inside want to get out. (adapted from Fortress Besieged)
For this quote to work, it is crucial to contrast between people that are outside and inside the fortress. Also, monosyllabic words tend to compound with localisers in modern Chinese (e.g. 桌上, 窗外), so that is another point of consideration.
Case 3: Precision is optional
住在大城市的人普遍比較有錢。 Those living in big cities are generally wealthier.
住在大城市裡的人普遍比較有錢。 Those living inside big cities are generally wealthier.
The first sentence sounds like a general description. Some may think 裡 in the second sentence is redundant, but it's not as wrong as the example sentence in Case 1. Because technically, 'people of big cities' are 'people living inside big cities'. So precision here is optional.
With these in mind, let us look at your sentences.
我剛到美國的時候只能在餐館的廚房裡洗碗賺錢, then localising the kitchen with 裡 is perhaps more justified than localising the restaurant itself.貼 (to paste) is a rather fine action. The object requires a localiser.門.It is rather easy to determine when to use 在, 上, 下, 中, 裡 through their English equivalents.
在 - in, at
在...上(下) - on top/bottom of something. 上 is "top", 下 is "bottom".
在...中 - in between something, or among several things (在中間, 在其中). 中 is "middle".
在...裡 - inside of something (在這裡, 在那裡). 裡 is "inside".
The first example doesn't need a localizer to specify it is 'in' a location
今年夏天我在北京学习中文。- it doesn't need a localizer to indicate it is in Beijing -- no one would think it is in front of or above the city
The second two examples also stating an unspecified location
我刚到美国的时候只能在餐馆打零工 - It can be any restaurant; no one would think it is in front of or above a restaurant, so a localizer is not needed
他不在餐馆打工了 - It can be any restaurant, and no one would think it is in front of or above the restaurant, so a localizer is not needed
The next three examples stating a specified location
我要在CSE上贴我的问题 - CSE is a specific website
If you wrote 我要在 Facebook 上贴我的问题. That would imply you are posting your question on your own Facebook page
If you wrote 我要在 Facebook 贴我的问题, then it could mean you are posting on anyone's page on Facebook
他写了一个便条在门上 - '上 (on)' in "writing a note on a door" is required to indicate it is on the surface specifically,
他貼了一个便条在门後 - '後 (behind)' in "posted a note behind the door" is required to indicate it is behind the door specifically
他藏了一張便条在门內 - '內 (inside)' in "he hid a note inside the door" is required to indicate it is inside the door specifically
商店里的服务员对顾客都非常客气 - If you wrote 商店的服务员对顾客都非常客气, it could mean all or any store
In conclusion: You need a localizer when you need to indicate a specific location (e.g. in, on, behind, or above a door); you don't need a localizer when you don't need to indicate a specific location (e.g. '在北京' can only mean 'in Beijing')
More examples:
'在北京上空出現 UFO' cannot be reduced to '在北京出現 UFO' (that would imply UFO is found in the city)
'在天空出現 UFO' doesn't need '裡' to indicate it is in the sky, but you can still write '在天空上出現 UFO' to stress it is 'the sky above' or '在天空中出現 UFO' to stress it is in the middle of the sky
