Can you install Laserfiche snapshot as a standalone installation without installing the entire Laserfiche client?
Yup. When you do a custom installation, simply check Snapshot and uncheck everything else. You can also do this in an unattended installation by only using SSADDLOCAL to indicate what to install.
Can you install Snapshot 9.1 on a machine/production environment using 9.0.1 then?
(Will it work properly?)
No, Snapshot is still part of the 9.1 client package and the 9.0 and 9.1 client packages cannot be installed on the same machine at the same time. The 9.1 client package will upgrade the 9.0 package to 9.1. If you had then chosen to just install Snapshot, the other components would be removed. That's just how windows installer works.
If you mean other 9.0.1 components besides the Client installation package, then sure.