How does bitcoin prevent this network splitting attack? How are malicious nodes prevented from, after getting the nonce of a new block found by someone else, broadcast it to the other nodes claiming it to be its?

Also, how does the bitcoin system solve the situation of an evil miner who has found the nonce of a block and then sends it with 2 different nodes, each one reaching 50% of the network, therefore having a problem of ownership of this work?

Finally, if all nodes have to check a broadcasted nonce to prove it's legit, how does the network defend against attacks in which thousands of ip adresses (e.g. botnet) start sending wrong nonces, causing a cpu DDoS because of all the peers hashing and checking these nonces.

Thanks for the answers.

Note: As the three questions are related I thought that putting them all in a single question was correct, if not told me and I will edit the text to only contain the first question, placing the other two in different questions.
How are malicious nodes prevented from, after getting the nonce of a
  new block found by someone else, broadcast it to the other nodes
  claiming it to be its?


The nonce is found against a block which contains the coinbase transaction which claims the reward to the address of the person who found that nonce. So you can't claim it to be yours. You're incentivized to broadcast it to as many peers as possible to help it spread more quickly, but that nonce only works for the block which contains that specific coinbase transaction. Changing the coinbase transaction to claim the coins yourself would result in a different hash of the block and therefore you'd need to find a new nonce.


  Also, how does the bitcoin system solve the situation of an evil miner
  who has found the nonce of a block and then sends it with 2 different
  nodes, each one reaching 50% of the network, therefore having a
  problem of ownership of this work?


Not sure I understand this question but if there's a node with more than 50% of the hashing power of the entire network combined, we're in for larger problems.


  Finally, if all nodes have to check a broadcasted nonce to prove it's
  legit, how does the network defend against attacks in which thousands
  of ip adresses (e.g. botnet) start sending wrong nonces, causing a cpu
  DDoS because of all the peers hashing and checking these nonces.


Again, nonces work with the hash of a block. All you have to do to check the validity of the nonce is see whether the resulting hash is smaller than a specific value. If it is, you've found a valid block! So there's no problem in checking or DDoSing the nonces. In fact that's what mining is: a race for finding a valid nonce for the next block.