How does a node find a transaction in the blockchain? If a (light) SPV client asks a full node for the Merkle branch containing a specific transaction (or the value of an address), how does the full node find this transaction (or the UTXOs) in the blockchain?
Does the full node scan linearly through the entire blockchain or is there a faster way to link transactions and blocks?
You can read the bip37 specification for all of the gritty details. 


  how does the full node find this transaction (or the UTXOs) in the blockchain?


The client builds a filter which contains what they're interested in, be it output script (addresses), public keys, or TXID. They then send this to the node they are connected to. When they request data for a particular block the node filters it to only contain things which match the clients bloom filter, and sends it to them. To find all of their unspent UTXO, they request every single block in the chain in turn and look for things that could be theirs. As you mostly worked out, this does involve for every client the node loading 80GB of blocks from disk and performing extremely expensive filtering operations on them. 

Modern nodes recognise how terrible of an idea this is, and allow users to disable the functionality completely due to the massive load and denial of service risk involved. With specially crafted transactions, a remote peer with a single command of less than 50 bytes can cause a node to load a 1MB block from disk, hash it over a million times, and then return nothing.