All of the SHA functions allow unlimited input size. And yes, bcrypt computation time dwarfs that of SHA-3.
The SHA-3 family has "extendable-output functions," which can ostensibly be used to generate unlimited numbers of bits (albeit with only a given security level). These are new to SHA-3.
SHA-3 has more internal state, it really is plausibly better at handling very large data. If 'unlimited' is really less than a gigabyte, there's no problem. It's mostly the preimage series of attacks and length extension at that point. SHA-3 is better on those. SHA-512 has zero length extension attack resistance.
Internal state length may be a bit of a red herring (note that SHA-3 makes up for that longer internal state by ingesting more data per round), but SHA-3 probably has a higher security margin than the SHA-2 construction mostly because we have had sponge constructions for less time than we have had Merkle-Damgard constructions. NIST basically forced a higher security margin on SHA-3. You are correct about the length extension attacks (although these are mitigated by using SHA-2-512/256 for example), but I don't think that matters here.
The SHA-3 family has "extendable-output functions," which can ostensibly be used to generate unlimited numbers of bits (albeit with only a given security level). These are new to SHA-3.