What prevents the currently picked mining node from only confirming half of all transactions he picks up?
I think you misunderstand mining. It is not the miner picked by the network, but miner tries to pick the lucky number that makes his block legit. Miner are free to include whatever subset of valid unconfirmed transactions they like. The next lucky miner will pick out of the leftover and new ones.
Or from packaging different versions of the block together and passing them to only half the network?
You also misunderstand peer-to-peer networking. You can not pass something to "half of the network". Any message sent will most likely reach the entire network. The sender has no limiting control on the propagation, he can only make it more likely to propagate by distributing it to more peers at his own.
