No. As long as both devices use the same iTunes Account, any re-purchase of an already purchased game is free. In other words, if you re-purchase a game that you have already payed, you will not be charged again.
You don't need to have an internet connection during the normal use of Boardgame Companion. However, you need a connection to perform the following operations:
purchase a game
share a played game via email
In particular, once you have purchased a game, you don't need a connection to use it.