Learn Hardcore Korean: obeune yachaereul guwo meogeoyo. - I roast vegetables in the oven and eat them.

오븐에서 would emphasize the place where the action happens (“at the oven”), which is possible but less common when you mean “into.”
오븐으로 would treat the oven as an instrument or means (“using the oven”). It’s also correct if you want to stress that you’re using the oven to cook:
• 오븐으로 야채를 구워 먹어요.
In short:
오븐에 = “into/in the oven” (location or destination)
오븐에서 = “at the oven” (action location)
오븐으로 = “with/by the oven” (instrument)

What does the structure 구워 먹어요 mean? Why not just say 굽어요 or 먹어요?

구워 먹다 is a compound pattern: verb stem + -아/어 먹다, meaning “do V in order to eat it” or “eat it after doing V.”

굽어요 simply means “I roast/bake (something).”
먹어요 alone means “I eat (something).”
구워 먹어요 emphasizes you roast it and then eat it (so you’re enjoying the freshly roasted vegetables).

Why does 굽다 change to 구워 in 구워 먹어요 instead of 굽어 먹어요?

굽다 is a ㅂ-irregular verb. When a verb stem ending in ㅂ is followed by a vowel (like -어), the ㅂ changes to 우.

What level of politeness is the verb ending -어요 in 구워 먹어요? How would I make it more formal or more casual?

The ending -어요 is informal polite (해요체).

Why is the subject omitted in 오븐에 야채를 구워 먹어요? What would the full sentence look like with a subject?

Korean often drops the subject when it’s obvious from context. Here, the speaker is understood to be “I” or “we.”
Full version:

Why is 야채 marked with 를 as in 야채를? Could it take 은 or 가 instead?

를 is the direct‐object particle, marking what you roast and eat.

야채가 구워 먹어요 → “The vegetables (not something else) are roasted and eaten.” (subject focus)
야채는 구워 먹어요 → “As for vegetables, I roast and eat them.” (topic contrast)
But if you’re simply saying “I roast vegetables,” use 야채를.

Can I replace 야채 with 채소? Are there any differences?

Yes. 야채 and 채소 both mean “vegetables.”

How do I say this sentence in the past tense or future tense?