Minecraft – Can Minecraft be played without being connected to the Internet

Really depends on your use case.

You only need one copy of the actual game installed if they are taking turns playing.

If you want them to play multiplayer, you will need to purchase two accounts, one for each kid. This also allows them to easily have their own name and "skin" in game.

However, if they just want their own names, you can create two shortcuts to the game, one for each kid. Then just give a different name for each shortcut. See this question on creating the shortcuts.

No, you need to at-least log in once to prove that you bought the game.