1. Yes, it should be "happily."
2. Comma after "majesty" not a semicolon.
There should be no question mark in #2 at all; however, that particular question mark is where you need to separate two sentences correctly. How will you do that?
1. Little would have lived very happy in this good family if it had not been for the ill-natured cook.
(happy-> happily...right?)
2. The captain, who knew his business, took this opportunity to set forth the merits of Miss Puss. He told his majesty; "It is not very convenient to part with her, as, when she is gone, the rats and mice may destroy the goods in the ship?but to oblige your majesty, I will fetch her."
(Would you correct some errors,plese? What's the use of '?" after 'ship'?)
3 answers
Do we have to use ':' instead of '?'?
No, it'd be better to write it this way:
He told his majesty, "It is not very convenient to part with her, as, when she is gone, the rats and mice may destroy the goods in the ship. However, to oblige your majesty, I will fetch her."
He told his majesty, "It is not very convenient to part with her, as, when she is gone, the rats and mice may destroy the goods in the ship. However, to oblige your majesty, I will fetch her."