Is there special code somewhere that keeps track of all the btc not spent in all the transactions of a block and allows the miner's address (ie the address in the first tx of that block) to spend that?

Something like that:
Every block has a coinbase transaction at the beginning.
The coinbase transaction is special, in that it can pay coins to its outputs without having to specify an input.

The total value of the outputs of the coinbase transaction can be at most the block value (the fixed amount per block) + the fees (summed over all transactions in the block).

This is checked in ConnectBlock in 
https://github.com/bitcoin/bitcoin/blob/master/src/main.cpp#L1711
The value of the coinbase transaction is being checked on line 1731.

But if so, how would that work then? I should be finding transactions then where the sum of the ouputs is higher than the sum of the inputs and one of the inputs should have the miner's address attached to it or what?

The only transactions in which the sum of the outputs can be larger than the sum of the inputs are the aforementioned coinbase transactions (which cannot have inputs).

The outputs from the coinbase transaction can be spent as any other output.

Wladimir