DIV Player for Android has a media library for video files, and allows to browse folders directly.

VLC has support for multi-track audio and subtitles. It supports auto-rotation, aspect-ratio adjustments and gestures to control volume, brightness and seeking.

It also includes a widget for audio control, supports audio headsets control, cover art and a complete audio media library.


Permissions
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VLC for Android needs access to those categories:
• "Photos/Media/Files" to read your all media files :)
• "Storage" to read your all media files on SD cards :)

Permission Details :
• It needs "read the contents of your USB storage", in to order to read your media files on it.
• It needs "modify or delete the contents of your USB storage", in order to allow deletion of files and store subtitles.