Manage FSx Storage Gateway Cache

Yes, For FSx type storage gateway you can set refresh cache based on TTL but you can't run manually that is only for S3 file type storage gateway.

Coming to your 2nd question, it is not possible to pin specific folders or files to cache. Storage gateway has built-in intelligence algorithm which manages what files to store in or evict from cache. However, all Files metadata only is always stored within storage gateway root disk.