How to merge multiple fields into one, across multiple content types?
By using Views (to build a list of nodes to be updated) and a combination of Rules and VBO (to mass update such lists) you should be able to perform the merge of those fields.

Refer to my answer to "How to bulk update select list fields?" for an example with way more details about that, which is about a variation of your scenario (ie change the value of an existing field, while you want to "copy" the value of an existing field to the value of a new field).

For your case, this example of a Rules Component (in Rules export format) should be close to one of the Rules Components you'd need for your "field_image":

{ "rules_set_an_initial_value_for_image_field" : {
    "LABEL" : "Set an initial value for image field",
    "PLUGIN" : "rule",
    "OWNER" : "rules",
    "REQUIRES" : [ "rules" ],
    "USES VARIABLES" : { "node" : { "label" : "Node", "type" : "node" } },
    "IF" : [
      { "entity_has_field" : { "entity" : [ "node" ], "field" : "field_article_image" } }
    ],
    "DO" : [
      { "data_set" : { "data" : [ "node:field-image" ], "value" : [ "field_article_image" ] } }
    ]
  }
}


Note: by using an approach with modules such as above (instead of SQL magic), you do have the advantage that all possible relatic magic will happen automagically and behind the scenes (eg: create a new revision of such nodes). And it avoids your Drupal database gets corrupted because "you forgot some table".