Sure, here are the corrections and explanations for each sentence:

1. **Original:** Our teacher gives us much homeworks today.
**Correction:** Our teacher gives us much homework today.
**Explanation:** "Homework" is an uncountable noun and does not have a plural form. Therefore, "homeworks" should be corrected to "homework."

2. **Original:** His sister often goes to school with an orange small bag.
**Correction:** His sister often goes to school with a small orange bag.
**Explanation:** In English, adjectives usually follow a specific order. The correct order in this case is size (small) before color (orange).

3. **Original:** John doesn&t work so hardly as Peter, but he is much more careful than Peter.
**Correction:** John doesn’t work as hard as Peter, but he is much more careful than Peter.
**Explanation:** The adverb "hardly" means "barely" or "scarcely," which is not the intended meaning here. The correct adverb is "hard."

4. **Original:** Do you know how much does it cost to fly to Australia?
**Correction:** Do you know how much it costs to fly to Australia?
**Explanation:** In indirect questions, the subject and verb do not invert. The correct form is "how much it costs."

5. **Original:** She’d like a dozen eggs and any vegetables.
**Correction:** She’d like a dozen eggs and some vegetables.
**Explanation:** "Any" is typically used in negative sentences and questions. In positive statements like this one, "some" is the appropriate choice.

I hope this helps!