Why can malicious miners not award themselves with any number of bitcoins? Why can malicious miners not award themselves any number of bitcoins? Is that because the award itself is a transaction which has to be confirmed by a random miner? And if so, what if the miner himself will win the race?

Also, why can't any miner that has solved the block broadcast a self-award transaction at any time?

Also, if the award is in a transaction, who is the sender?
A miner is at perfect liberty to create any sort of block he wants and broadcast it to the network.  This could include rewarding himself with an unlimited number of bitcoins, or including transactions without proper signatures.  However, this block means nothing if other miners do not consider the block valid.  If other miners reject the block, then it doesn't matter what the block says.

This creates a situation where a miner is wasting his valuable computational resources if he chooses to do the proof-of-work on an invalid block.  He would be throwing away the money invested into the mining hardware.  Unless he has reason to believe that the majority of hashing power on the network will accept his invalid block, it's in his best interest to follow the rules of the system.  A situation like this is known as a 51% attack.

So long as hashing power is not centralized into the hands of those colluding with one another, Bitcoin is considered safe from this sort of vulnerability.