2. This idea is ok. First of all, I expressed that there are some new books in the library, and I want to invite Alice to have a look together. If she is inconvenient, she should call her and tell her. I hope Alice can reply as soon as possible
Recently, I got news that some new books in the city library have been listed in the catalogue (I want to express that there are more books in their library, can I? ), which is very useful for us to have grammar problems in English learning. I got news recently that there are some new books in the city library that are very useful for our English study. This expression is concise and clear.
How about going to the library with me this weekend? How about going to the library with me this week? When expressing invitations, most people use what about.
4. If it is inconvenient for you, it is usually understood as unavailable, and should be changed to unavailable. Obedience is generally used to express the convenience of objective conditions.
Finally, I hope you can reply to me as soon as possible. An inappropriate reaction refers to a reaction to an event. This should be changed to reply.