Is there a tool or equivalent algorithm for the logical fourth one: "Merge Parts"?
No, there isn't.
The adjacent parts of a multipolygon are considered a topological error and must be linked with the Fix geometries tool.
Is it even possible to select individual parts?
No, it isn't.
You can select features, not individual parts.
Is there anything else I'm missing?
I don't know. But you can refer to parts of collections with the geometry_n()
function in expressions.
For example, you can create a new feature (in a separate layer) with the Geometry by expression tool, that contains the combination of two parts (or more if the function is nested) of a feature or of several, of any layer. If the parts are adjacent, they will form a single part in the new feature.
To create a feature formed by parts 1 and 2, of the feature_ID 0, of the 'multipolygons' layer:
combine(
geometry_n( geometry( get_feature_by_id( 'multipolygons', 0)), 1),
geometry_n( geometry( get_feature_by_id( 'multipolygons', 0)), 2)
)
Note than the Geometry by expression tool copies the table from the input layer. If the input layer have more than one features, more than one features will be created, but only one will have a valid geometry with that expression.
UPDATE:
About the WKT of the Multipolygon provided in the comments (https://pastebin.com/1ibYDzR5), I can't seem a line shared by both parts.
When loading the feature in QGIS, it can be seen (with an appropriate zoom level), that the parts are not adjacent.
Upper:
Lower:
Be sure to snapping when digitizing and editing (here is the manual). Otherwise, you can not merge those parts as they are in any way.
Best Answer
Inspired by this question & answer as an example of how "easily" can one make its own solutions when using Open Source, I have tried to create my own code to selectively "explode" multipart features during an editing session.
I have explored the QGIS 1.8 API for the first time, and came out with this piece of code that seams to do the job:
I'm not an experienced programmer, so the code might not be the most efficient one.
Next steep will be to make a plugin out of it... Or at least, try to!
UPDATE:
I was able to create the plugin. It's called Multipart Split, and can be found in the QGIS official repository.