Do native speakers think about compound words as "东西"?
When I saw a tomato first in my life time, I was told its name was fanqie(番茄) by my mom or dad in the dialect. I was not able to recognize or write any character then. All I knew about the tomato was that it was a red, green, or yellow, and delicious vegetable and sounded fan qie. I thought it was another kind of 茄子(eggplant).

Later, after I went to school and learned to speak Putonghua and write characters, I got to know that fan qie is written as 番茄 and it has another name 西红柿. As my teacher explained, 番 refers to foreign. 番茄 is a kind of eggplant-like thing from a foreign land. The name makes sense. So does 西红柿, western land's red persimmon. But now when it comes to 西红柿 or 番茄, I don't think about what each character means. It's just a name of an object. 

In some cases, I may not really care about each character's meaning from the very beginning. For example, 深圳. It's a city of Guangdong province. If 圳 alone is seen, I may not recognize it at all. Seldom can I see it elsewhere. It's always part of 深圳. So I don't know its meaning till now without referring to a dictionary, and I'm sure it won't take long before I totally forget it even if I look it up.

Another example is a transliterated word like 咖啡. I didn't figure out what 咖 or 啡 means until I learned the English word coffee. And all of a sudden, I got to know there are many such words in Chinese, like 坦克(tank), 沙发(sofa). A single character in such word has no specific meaning but just indicates the pronunciation. Of course there are some translated words that concern about both the meaning and pronounciation like 迷你(mini), 可口可乐(Coco Cola), 露华浓(Revlon) and etc, some of which are really beautiful and elegant master pieces of translation.

A third example is a word like 麒麟, kirin. I thought it was just a creature in myth, and didn't try to find out what 麒 or 麟 means until I watched an anime， the Twelve Kingdoms, in which there are kirins that can transform to humans. The female kirins are named X麟 and the male X麒, where X is a character that has similar pronounciation with the name of the kingdom the kirin belongs to. After some research I leaned that 麒 refers to a male kirin and 麟 a female kirin. So does 凤凰, phoenix, 凤 as male and 凰 as female. But in the word 龙凤, 凤 represents a girl or woman.

There are some characters in my dialect I have been expecting to find out how to write. Some have been clear, but the others have not. There is a dirty character that means the f word. It sounded exactly the same with 十(ten) in the dialect. Since it's a dirty word, I just didn't dare and felt embarrassed to ask my parents or teachers how to write it correctly. My little fellows and I just wrote it as 十. In the circumstances, we all knew it meant the f word. About one year ago, I happened to read someone's post and get to know the right character should be 入(go into, penetrate). A 20-year-old question was finally made clear. However some of the unclear may stay mysterious till my last day. 

I believe every character in a compound word makes sense, indicating either the meaning or the pronounciation or both. Now I always tend to find out what each character means when I see a new word . And I would do the same to the old words, expecting to find some interesting stories. It helps to get a deeper understanding and also bring benefits especially when learning ancient Chinese.