我写得对吗?

2. 他正在听着音乐骑自行车，他很不注意安全！

— 他听着音乐骑自行车: He listened to music while he was riding a bike.

‘骑自行车’ is a primary action in this sentence. If you want to describe another action at the same time, you should add ‘着’ behind the action.

— ‘不小心’ is an adv. It usually means ‘accidentally’

他不小心摔了: He fell down accidentally.

— ‘不注意安全‘ means someone do not care about his safety.

3. 在美国工作被做完了，他正在回国。

— if you want to use ‘被‘, it would be better to explain that the job was done BY WHOM.

— 他正在回: he is coming back.

—他回了: he came back.

4. 服务员说着“欢迎”给我们新的菜单。

I don’t think this sentence is incorrect. But please notice the punctuation. 《》is only used for a book’s name.

5. 我觉得她正在世界上旅游，没有？

The sentence is correct, but the meaning is weird. ‘世界上‘ means ‘be in the world’ (be, exist). You might mean to say ‘ around the world’, which you can use ‘周游世界‘

— 我觉得她正在周游世界，不是吗？

I think she is traveling around the world, isn’t she?

我写得对吗?

在美国工作被做完了，他在回国了。

 在美国，工作结束后，他回国了。

服务员说着《欢迎》给我们新的菜单。

服务员一边说着《欢迎》一边给我们新的菜单。

我觉得她正在世界上旅游，没有？

我觉得她正在环游世界，对吗？ 

I hope my correction  helps.

If you have any questions about Chinese,feel free to massage me. Cause I provide correcting  service for the learner who has on advanced level on italki. And I also teach the basic Chinese you can have more details in my profile if you click my portrait.^ ^