Learn Hardcore German: Gute Teamarbeit macht das Projekt einfach.

Teamarbeit is a feminine noun in German (like die Teamarbeit in the dictionary) and here it is in the nominative singular as the subject of the sentence.

For a feminine noun in the nominative singular with no article, the adjective ending is -e:

gute Teamarbeit – correct
gut Teamarbeit – wrong (missing the adjective ending)
guter Teamarbeit – would be genitive or dative, not nominative

Pattern (no article, feminine, nominative):
gute Frau, rote Rose, gute Teamarbeit.

What exactly is the subject of the sentence?

The subject is Gute Teamarbeit.

Gute = adjective describing what kind of teamwork
Teamarbeit = the noun (teamwork)

So the structure is:

Subject: Gute Teamarbeit
Verb: macht
Object: das Projekt
Predicative adjective (resulting state): einfach

Meaning: Good teamwork makes the project easy.

Is Teamarbeit singular or plural, and what gender is it?

Teamarbeit is:

Singular in this sentence (it means teamwork as an uncountable concept)
Feminine gender in German: die Teamarbeit

There is no plural normally used for Teamarbeit, just like teamwork in English is not usually plural.

What case is das Projekt, and how can I recognize it?

Das Projekt is in the accusative case as the direct object of macht.

You can see that from the structure:

Subject: Gute Teamarbeit (who/what does something?)
Verb: macht (does what?)
Direct object: das Projekt (what is being made into something?)

For neuter nouns, der/das/dem/den patterns are:

Nominative: das Projekt
Accusative: das Projekt
Dative: dem Projekt
Genitive: des Projekts

Nominative and accusative look the same for neuter, so you identify the case by function in the sentence: after machen (to make), the thing being made is normally accusative.

Why is it das Projekt and not der Projekt or die Projekt?

In German, every noun has a fixed grammatical gender that you just have to learn with the word.

Projekt is a neuter noun: das Projekt
If it were masculine, it would be der Projekt
If it were feminine, it would be die Projekt

But the correct dictionary entry is das Projekt. This doesn’t change in the nominative or accusative singular: it stays das in both cases.

Why is the verb macht used and not ist? Could I say Gute Teamarbeit ist das Projekt einfach?

No, Gute Teamarbeit ist das Projekt einfach is not correct German.

machen here has the meaning to make something become / be in a certain state, just like in English to make the project easy.
The pattern is:
[Subject] macht [object] [adjective].
→ Gute Teamarbeit macht das Projekt einfach.

If you use sein (ist), the pattern is different:

Das Projekt ist einfach. – The project is easy.

That sentence just describes the project; it doesn’t say what causes it to be easy. Machen expresses that good teamwork causes the project to be easy.

Can I also say Gute Teamarbeit macht das Projekt einfacher or leichter? What’s the difference from einfach?

Yes, you can say:

Gute Teamarbeit macht das Projekt einfacher.
Gute Teamarbeit macht das Projekt leichter.

Differences:

einfach – easy/simple as an absolute description.
Gute Teamarbeit macht das Projekt einfach.
→ Teamwork makes the project (generally) easy.
einfacher – easier, a comparative form.
Gute Teamarbeit macht das Projekt einfacher.
→ Good teamwork makes the project easier (than it would otherwise be).
leichter – also easier/lighter; very similar to einfacher in this context, with a shade of meaning like less effort, less burden.

So einfacher/leichter focus on comparison; einfach just states that the project is easy as a result.

Where does einfach belong in the sentence? Could I say Gute Teamarbeit macht einfach das Projekt?

In this sentence, einfach is a predicative adjective describing das Projekt after machen, so it belongs at the end:

Gute Teamarbeit macht das Projekt einfach. – correct

Gute Teamarbeit macht einfach das Projekt is wrong in this meaning because:

There, einfach would be read as a sentence adverb, something like simply / just in English, and the sentence is still incomplete: Good teamwork simply makes the project… (…what?).

Correct patterns:

Gute Teamarbeit macht das Projekt einfach.
Gute Teamarbeit macht das Projekt sehr einfach.
Gute Teamarbeit kann ein schwieriges Projekt einfach machen.

Can I drop the article and say Gute Teamarbeit macht Projekt einfach?

In standard German, you cannot drop the article here. You need:

Gute Teamarbeit macht das Projekt einfach.

German usually requires an article with countable singular nouns like Projekt, unless you have another determiner (like dieses Projekt, jedes Projekt, etc.).

Without an article, Projekt sounds wrong or at best very telegraphic/heading-style, not like normal spoken or written German.

Why are Teamarbeit and Projekt capitalized?

In German, all nouns are capitalized, regardless of where they appear in the sentence.

Teamarbeit – noun → capitalized
Projekt – noun → capitalized

Adjectives like gute and einfach are not capitalized here (unless they begin the sentence or are part of a proper name).

Could I say Gute Teamarbeit vereinfacht das Projekt instead? Is that the same meaning?

Yes, Gute Teamarbeit vereinfacht das Projekt is grammatically correct and very natural.

Differences in nuance:

macht das Projekt einfach – literally makes the project easy; sounds a bit more informal/direct.
vereinfacht das Projekt – literally simplifies the project; a bit more formal or technical, focusing on the process of making it simpler.

In most everyday contexts, they are close in meaning and both would be understood as good teamwork makes the project easier/simpler.