Adding (and exporting) tags to pictures
Is there some application that can help me tag and organize pictures, 
  storing the metadata correctly in the photos themselves?


I do not know any android app that can do this.

But maybe you can help implementing this feature

Unfortunately the android internal media database/media scanner does not support "tags" for photos
so an android app has to implement the tag scanning/management on its own.

I also donot know a ready to use android java library that can write jpg iptc-tag infos.

currently i am thinking about adding tag-support to https://github.com/k3b/AndroFotoFinder/
using a pc generated csv file to transfer tags 

For details see https://github.com/k3b/AndroFotoFinder/issues/25

[Update 2016-11-09]

Implementation is under way using a seperate xmp - sidecar file for every jpg to store the iptc-tags.

There is much work ahead for this hobby project.

On the pc side you have to use an exiftools based batchfile to transfer between non-android-standard-jpg-fields and xmp-sidecarfile/csv

[Update 2016-11-09]

Implementation is done :-) with tag write support in jpg-exif or external xmp file