How could we implement a way that ICAP addresses can be used in the IBAN field when making a normal bank transfer? What risks would it bring?
I almost clicked "vote to close this question due to it being too broad". But let's try this way... Here's a reference to ICAP (had to google it also myself): https://github.com/ethereum/wiki/wiki/Inter-exchange-Client-Address-Protocol-(ICAP)
Short version: a lot of things and not going to happen in any near future.
Longer version of what would be needed:
I assume your scenario means that banks would remain as untouched as possible. For example they would not have Ether accounts for customers. So the banks would need internal Ether balance to exchange your fiat transfer into Ether and send the Ether inside the blockchain
Due to owning Ethers banks would need whole new kind of risk management systems: what if Ether goes up or down in value while they hold it?
Somebody has to fight the regulators. Banks have very strict rules about how they can transfer assets and sending assets to a random Ethereum address is certainly not within their possibilities currently.
Related to the previous point: banks would need whole new systems of identifying address holders, performing KYC & AML tasks etc.
Banks would need to run their own Ethereum clients (not a big problem)
A lot of other things which I can't think of right now