In this answer, I'll use your term (adjective), although I think it would be more proper to say attributive modifier.
When you have an article + adjective + noun sequence, the article relates to the noun, not the adjective. You decide whether to use the article based on whether it makes sense for the noun. In other words, to decide whether to use an article, remove the adjective and see whether an article is required:
- *I am studying checklist for the test on Monday.
- I am studying a checklist for the test on Monday.
- I am studying the checklist for the test on Monday.
Option 1 is ungrammatical. Options 2 and 3 both work, depending on whether you want to use a definite or an indefinite article. So, put your adjective back in:
- *I am studying SAT checklist for the test on Monday.
- I am studying an SAT checklist for the test on Monday.
- I am studying the SAT checklist for the test on Monday.
Once again, option 1 is ungrammatical, and options 2 and 3 both work. Please note that a becomes an, because SAT (pronounced /ˈɛs eɪ ˈtiː/) begins with a vowel sound.
Now for the question of a(n) versus the. Again, this relates to the noun, not the adjective. You decide which article to use based on the noun. If the noun were SAT, we'd probably say the SAT, because it's a specific test; we very rarely or never say an SAT for that reason. But in this case, the noun is checklist! It doesn't matter that SAT is being used as an adjective here--we can ignore it entirely when we're choosing which article to use. So again, let's remove the adjective:
- I am studying a checklist for the test on Monday.
- I am studying the checklist for the test on Monday.
Since this is the first time you've mentioned the checklist (it's not part of the conceptual space that is shared by the speaker and listener), we mark it as new information with the indefinite article a(n). Option 1 is better. Let's put the adjective back in:
I am studying an SAT checklist for the test on Monday.
And there we have it. Note that once again, a(n) is realized as an before a vowel sound.
In this answer, the * symbol marks a sentence as ungrammatical.