Here are the corrections for each sentence along with explanations:

1. **Incorrect**: My mother don’t went shopping last Sunday.
**Correct**: My mother didn’t go shopping last Sunday.
**Explanation**: The auxiliary verb "do" must be used in the past form "didn't" with the base form of the verb "go" (not "went").

2. **Incorrect**: My mother buyed lots of vegetable at this market.
**Correct**: My mother bought lots of vegetables at this market.
**Explanation**: The past tense of "buy" is "bought," and "vegetable" should be pluralized to "vegetables."

3. **Incorrect**: The storm did go into my town a few hours ago.
**Correct**: The storm did not go into my town a few hours ago.
**Explanation**: Since the sentence implies that the storm did not enter the town, it needs to be in the negative form.

4. **Incorrect**: I were not brave enough to face the problem.
**Correct**: I was not brave enough to face the problem.
**Explanation**: The correct form of "to be" in the past for the first person singular (I) is "was," not "were."

5. **Incorrect**: The orchestra begun playing at the concert.
**Correct**: The orchestra began playing at the concert.
**Explanation**: The past tense of "begin" is "began," not "begun."

6. **Incorrect**: I got a good mark at the final exam in the next week.
**Correct**: I will get a good mark on the final exam next week.
**Explanation**: "Next week" refers to the future; thus, the future tense is required. The preposition "on" is used with "mark."

7. **Incorrect**: It rain heavily during the day off.
**Correct**: It rained heavily during the day off.
**Explanation**: The verb "rain" needs to be in the past tense "rained" since it refers to a completed action.

8. **Incorrect**: They didn’t considere who is more talented.
**Correct**: They didn’t consider who is more talented.
**Explanation**: The correct spelling of the past tense of "consider" is "considered," but since it’s in the negative form here, it remains "consider."

9. **Incorrect**: Do you go out last night?
**Correct**: Did you go out last night?
**Explanation**: To ask about a completed action in the past, "do" should be changed to "did."

10. **Incorrect**: I crossed a few minutes the road to catch my cat.
**Correct**: I crossed the road for a few minutes to catch my cat.
**Explanation**: The sentence should clearly convey that the speaker crossed the road, and the phrase "for a few minutes" denotes the time spent.

In summary, here are the corrected sentences:

1. My mother didn’t go shopping last Sunday.
2. My mother bought lots of vegetables at this market.
3. The storm did not go into my town a few hours ago.
4. I was not brave enough to face the problem.
5. The orchestra began playing at the concert.
6. I will get a good mark on the final exam next week.
7. It rained heavily during the day off.
8. They didn’t consider who is more talented.
9. Did you go out last night?
10. I crossed the road for a few minutes to catch my cat.