We manage keys, but we don't store data.
All crypto operations and encrypted data processing happens inside TEEs (AWS Nitro Enclaves, specifically [0]). Using Relay, you can pass data on to trusted third parties over TLS. With Cages, you can deploy custom code inside a TEE which can process data in whichever way you need.
For developers who don't want plaintext data on our infrastructure, we also provide SDKs which let them encrypt data using our PKI scheme — on their own infrastructure.
[0]: https://press.aboutamazon.com/news-releases/news-release-det...
Here's my PSI project if interested: https://github.com/dowensagain/EfficientMultiPartyPSI
It is unlikely (in my opinion) that "true" FHE applications will be deployed any time soon, but "leveled" FHE applications are already being deployed for a small number of levels (e.g. 2). Beyond quartic functions the performance is probably going to be too much of a problem for most applications. Homomorphic encryption in general is commonly used as a building block in larger MPC systems and you will probably see more widespread use of leveled FHE as such a building block too.
As for selectively decrypting outputs, that sounds like functional encryption and it is still an active area of research (see also obfuscation, which was a hot topic a few years ago). I doubt you will see practical applications for a very long time.