As to faking signatures and, e.g. stealing Satoshi's coins or just fucking up the network with fake transactions that verify, there is some concern and there are some attack vectors that work well if you have a large, fast quantum computer and want to ninja in. Essentially you need something that can crack a 256 bit ECDSA key before a block that includes a recently released public key can be inverted. That's definitely out of the reach of anyone right now, much less persistent threat actors, much less hacker hobbyists.

But it won't always be. The current state of the art plan would be to transition to a quantum-resistant UTXO format, and I would imagine, knowing how Bitcoin has managed itself so far, that will be a well-considered, very safe, multi-year process, and it will happen with plenty of time.