Nope, the mechanics simply did not install the water pump properly. There is absolutely zero reasons why it would leak other than it was incorrectly installed.
absolutely not, take it back to them and demand they replace it again as it is defective or the gasket was not cleanly removed and a new gasket installed properly
No - should be bone dry. I'm not familiar with the Honda engine - but - I would worry about a leaking pump also ruining the timing belt which often happens.
No, if it is leaking it is defective.
Is It possible for a water pump to leak for a little bit when it's first replaced? It was installed professionaly and it's a 2002 civic ex 5 speed