Custom fields from media module not asked for when saving image

I may have misunderstood your question, please let me know if this is the case.

The entity embed module on its own is intended to allow you to select an existing entity and embed it inside a wysiwyg area. When used in conjunction with entity_browser (or any variation of it, as media_entity_browser), you could potentially have a workflow where entities can be created on the fly just before being embedded. If you are looking for this second scenario and are not seeing the entity fields in the entity creation form, your browser may not be configured properly.

On the other hand, if what you are looking for is a way of selecting an existing entity and then modifying some of its fields right before embedding it, this is not something possible with these modules. Actually, bear in mind that what is embedded is not the rendered markup of the entity, but only some metadata pointing to it so drupal can re-render it as many times as necessary in the future. Which means that even if you could modify some of its fields (using some custom code or other contrib solutions), this modification would affect all other instances where this entity is being used, either embedded in wysiwygs or referenced in other ways.

By the way, #2513086: [PP-2] Right click option to edit embedded entity: not reference a different entity, but *edit* the entity in a dialog may give you some patches to play around if that is something you are interested into.