If you lose Magic permanently (due to implants, for example), who chooses what Adept Powers are lost? What if there is no combination of powers which adds up exactly to an integer, or for whatever reason (player choice, GM fiat) you have points left over due to a lost power which cost more than the Magic you lost? Can you choose new powers with the point fraction left over, or save them for later?

The player gets to choose. If you have an arithmetic problem, you need to take at least enough points to cover the loss. Any extra can be used or saved. I guess it would be up to the GM how quickly you could convert the left-over points if you wanted to use them right away.