First, with the "Frost Bridge" advanced action card, how much is need to spend in move for cross a lake ?
It says on the card: the move cost of Lakes and Swamps is reduced to 1, so it's 1 point of movement to move onto a lake.
Second, with the the green mana power of concentration action card, if i play improvisation i need to discard a card too ? And with the same green power, can i play advanced action card ?
Yes, you must discard a card. Concentration allows you to use the powerful effect without spending the usual mana, but you must still spend any other costs associated with the card. And yes, you can use Concentration with Advanced Actions as well as Basic ones.