How can I manually download the Play Store APK from Google? I have a Xiaomi Mi Note that doesn't come installed with Play Store. I'd like to install an APK for it manually, but all resources I've found on the web require me to install a binary from untrusted sources (e.g., a Mediafile upload).

How can I download the Play Store APK directly from Google?
Google does not allow downloading the Play Store directly from the Play Store itself, unless its an update to an existing installation. What you can do however is:

1) Root your phone (Xiaomi warranty isn't voided by rooting)

2) Use an app like Flashify or fastboot to flash a custom recovery (CWM or TWRP) (Refer this guide).

3) Flash a compatible GAPPs package, which includes the Play Store APK. You can get it that way.