Forgive my ignorance, but if the address (Satoshi's genesis address) contains the unspendable BTC, does is mean other Bitcoins that get there become unspendable as well?
No. The coinbase reward from the genesis block is non-spendable. Any other outputs coming from that address can be normally spent.
If you want to "burn" some bitcoin you can generate an address, send the money there and ELIMINATE the private key that has access to it.
That won't be true burning either. First things first, nobody can assure you there won't be another fellow who'll generate the same key. Second, there are about 2^96 different keys, all of which result to the same address. If you want to burn coins, you need to remove them from the UTXO set, using
OP_RETURN.