Good job on your sentences!~ ^^ Keep it up as doing so can greatly enhance your vocabulary and grammar. ^^ 

*Correction: 

I don't like staff dinner every month.

--> I don't like having a staff dinner every month. 

His complexion is always bad in the morning.

--> Good sentence.~ ^^ 

By her pale lips, it seems like she swam for a long time.

--> She must have swum for a long period, judging by her pale lips.

Our team is finding the venue for the workshop.

--> Good sentence.~ ^^ 

Everyone is expecting whatever he shows.

--> Good sentence.~ ^^ 

I have terrible hangover I even drink a little.

--> I have a bad hangover and have just had a small amount of alcohol.

Don't give up our relationship easily.

--> Don't give up on our relationship easily.

They fought throughout late week, and they eventually split up yesterday.

--> They fought all through the week, and they eventually split up yesterday. 

Don't mention me on your mouth.

--> Don't bring me up in conversation.

I like a man have intelligent image.

--> I appreciate the image of a man who is intelligent.

This rapper not only humorous but also intelligent.

--> This rapper is not humorous, but intelligent as well. 

This architect is popular as building this office.

--> This architect is well-known for designing this office.