Sorry, forget AD797, it's only rated for high voltage dual supplies and isn't rail to rail. I missed your 5V supply requirement.
The good news is that your noise spec isn't really demanding by audio frequency standards.
Except for low frequencies where 1/f noise dominates and very high frequencies where feedback factor is insufficient to correct noise of gain/output stages, opamp noise is generally determined by the input stage, flat with frequency and rated in nV per root of Hz. You multiply the nV/rtHz datasheet rating by square root of bandwidth and get RMS voltage of noise.
The super-lousy jellybean LM358 is 40nV/rtHz, giving a few µV RMS over 20kHz bandwidth. Add some more because you will also amplify out-of-band noise due to imperfect feedback filters. That's still perfectly within your demands, it seems. And that's the lousiest, noisiest opamp you could possibly buy (not that you should, it's way too slow

).
Current noise could become a concern with bipolar opamps only if source impedance is more than several kΩ. Is it that bad?
So it seems your main points of concern are R-R output and enough bandwidth. Perhaps distortion, you said nothing about that.
I really have zero clue about the kind of opamps fast enough to do all of that in one stage. But you can get a dual and make two stages of 30~33x gain, particularly if gain accuracy isn't critical. Out of "audio" parts, perhaps OPA1612 or AD8397. I'm sure they are both overkill and some cheap-ass 200~500MHz chip may exists which could be even better, but I'm completely unfamiliar with that market. Parametric search is your friend, or wait for other folks' suggestions.
As for slew rate to reach 5Vpp in a few µs period signal, I think a few V/µs will suffice, which is typically met by any fast opamp

edit
Actually, AD8397 is a "video" opamp, although some people use it for audio too because of supposedly decent performance. Which perhaps is a good hint: look for analog video opamps. These are fast and quite accurate at high-kHz to low-MHz frequencies.