Learn Hardcore Korean: chaeksangeseo sukjereul haeyo.

The particle 에서 marks the location where an action takes place. Here, 책상에서 means “at (on) the desk,” indicating the place where you do your homework.

Why is 에서 used instead of 에 in this sentence?

에 often indicates a static location (“to be at” or “go to”).
에서 is used with action verbs to show where the action happens.
Since 해요 (“do”) is an action, 에서 is correct.

What kind of verb is 해요, and what speech level does it represent?

해요 is the present tense of 하다 (“to do”).
It’s in the polite informal style (해요체), which you use with people you’re not extremely close to but still want to sound friendly.

Why is 숙제 followed by the object marker 를 in 숙제를 해요, and can you omit it?

를 marks 숙제 as the direct object of 하다.
In formal or written Korean you usually keep it. In casual spoken Korean you can drop particles for brevity:
“책상에서 숙제 해” is common in conversation.

How would you change the sentence to the past or future tense?

Past tense (did): 책상에서 숙제를 했어요.
Future tense (will do): 책상에서 숙제를 할 거예요.

Can you switch the order of 책상에서 and 숙제를? For example, 숙제를 책상에서 해요?

Yes. Korean word order is relatively flexible. Both 책상에서 숙제를 해요 and 숙제를 책상에서 해요 mean the same thing. The first element often just sets the topic or emphasis.

What’s the nuance difference between 책상에서 and 책상 위에서?

책상에서 simply means “at the desk” (location of action).
책상 위에서 literally “on top of the desk,” emphasizing the surface.
In many contexts 책상에서 is enough, but 위에서 adds a bit more precision.

How do you form a negative sentence like “I don’t do homework at the desk”?

You can use 안 or 지 않다 before the verb:
1) 책상에서 숙제를 안 해요.
2) 책상에서 숙제를 하지 않아요.