A part of something literally means that it is just one part in a bigger aggregate structure among all other parts in that structure. The head is a part of the human body. It is one part out of many other parts that constitute the human body.
When something is part of something else (no article), it means that that something belongs to that something else or is found there. Since I'm a very peaceful person, violence is not part of what I am.
To sum up what I have just said, I think of a part of something as being physical and part of something as something more abstract, but in your case, you would probably be better off using the part of because the part that we are talking about is identified by additional information provided further down in the sentence.
Every lesson helps me understand American culture and its seemingly easy language.
I see no reason why you'd have to substitute "straightforward" for "easy" in that sentence. You could use "straightforward," but I think "easy" works just as well. There might be some subtle difference in the shades of meaning between the two, but I'm not sure that's especially important here. If "easy" is what you want to say, go ahead and say it.
And yes, I think the sentence as a whole sounds natural.
One thing that might help you get your idea across a little better is if you added a bit of detail around "seemingly easy." A native speaker won't likely have much trouble inferring from your wording that you don't think English is actually all that easy, but without including some idea to set up the inclusion of "seemingly," it does sound slightly out of place to my ear.
You could make your idea more clear by saying something like
Every lesson helps me understand American culture and the nuances of its seemingly easy language.
That way, you're not saying that they help you understand the language as a whole while also randomly implying that you think it's not an easy language, but rather you're saying that they help you understand the parts of English that aren't all that easy.
The revision certainly isn't strictly necessary, but I think it makes your compliment sound even more complimentary.
Best Answer
Whether to avoid tautology or not is your personal preference but "summary" by definition is "a brief statement or restatement of main points, especially as a conclusion to a work". You could surely say "brief/short/compact/condensed summary" but that would be buttered butter.
Better to use words like "description", "note", "article", or possibly "extract" with such adjectives. Notice that adjectives like "concise" or "succinct" can work with "summary" since besides "briefness" they mean "being comprehensive" or otherwise "giving a lot of information clearly".