Here are the sentences with the verbs in the correct tense, along with explanations:

1. It is very hot. Summer **has come**.
- (Present Perfect): "has come" emphasizes the result or the current situation stemming from summer arriving.

2. They **have studied** at this school for 7 years now.
- (Present Perfect): This indicates an action that started in the past and continues to the present.

3. They **began** to study at this school 7 years ago.
- (Simple Past): This specifies a completed action that occurred at a specific time in the past.

4. Since **when did you know** her?
- (Simple Past): The question refers to a specific point in the past when the speaker first became acquainted with her.

5. How long ago **did he start** to learn French?
- (Simple Past): This refers to a specific event in the past, asking for the time since it happened.

6. She **lost** her handbag on the train yesterday morning.
- (Simple Past): This refers to a specific event that happened in the past.

7. My mother **has been** in hospital for a long time and she cannot go home yet.
- (Present Perfect): This emphasizes that the action started in the past and continues into the present.

8. His dog **has just run** out of the garden.
- (Present Perfect): This indicates an action that has recently occurred and has relevance to the current situation.

9. Mr. Brown **has traveled** by air several times in the past.
- (Present Perfect): This highlights experiences up to the present moment.

10. He **traveled** to Mexico by air last summer.
- (Simple Past): This statement refers to a completed action that took place at a specific time in the past.

### Summary of Tenses Used:
- **Present Perfect** is typically used for actions that have relevance to the present or continue into the present.
- **Simple Past** is used for events that occurred at a specific time in the past and are completed.