Is the transaction within Genesis block spendable? Are the transactions added to the genesis block spendable, just like any other transactions from other blocks, or is that an unspendable coinbase transaction.

In other words..

Is the transaction within the Genesis block spendable, provided the corresponding private key of the bitcoin recipient is found?

and If I fork the Bitcoin code, and generate my own genesis block. Can I use the private key/public key I used as bitcoin recipient of the transactions  hard-coded with the genesis block?
Are the transactions added to the genesis block spendable, just like any other transactions from other blocks, or is that an unspendable coinbase transaction.


Transactions aren't spendable; transaction outputs are.


  Is the transaction within the Genesis block spendable, provided the corresponding private key of the bitcoin recipient is found?


The single transaction output in Bitcoin's genesis block is not spendable. It is not known whether this was an intentional choice by the system's creator, or an accident. Regardless, the reasons why currently don't matter, as no Bitcoin full node software would accept such a spend.


  If I fork the Bitcoin code, and generate my own genesis block. Can I use the private key/public key I used as bitcoin recipient of the transactions hard-coded with the genesis block?


In your own software you can enforce whatever rules you like.