Mongodb array vs object

I would store each guess as a separate document in a user guesses collection. The structure of each document would be as follows:

Guess
- userId
- selectedImageId
- correctImageId

You may also store more information about the corresponding quick, such as the shown images and played audio. To generate statistics, you will need to run Map/Reduce over this collection. For example, to get stats of total and correct guesses, your Map/Reduce output document structure would be:

UserStats
- userId
- guesses
- correct

The map function may look like this:

function() {
    emit(this.userId, { guesses: 1, correct: this.selectedImageId === this.correctImageId ? 1 : 0 });
}

And the reduce function:

function(key, values) {
  var result = { userId = key, guesses: 0, correct: 0 };
  values.forEach(function(value) {
    result.guesses += value.guesses;
    result.correct += value.correct;   
  });
  return result;
}

Note, on MongoDB, Map/Reduce is not run automatically as source collections are updated, so you will need to implement that on your own.

Store hashtags in an array within a document.

That's the benefit of having documents: you can simply nest them. And, in this particular case, it's trivial:

{
    "_id": 123,
    "file": "c43a5f46-kitten.png",
    "description": "My kitten :3 #kittens #cute"
    "hashtags": ["kittens", "cute", "cat", "animals"]
}

(I added some "synonymous" tags, this can be done automatically by looking up some other document.)

This is the most natural solution for document-oriented database:

• Searching documents by hashtags is trivial if you just add an index, as well as inserting, updating, and deleting hashtags on random documents is also trivial
• Massive inserting, updating, and deleting is a bit tricky, because you'd probably want to split such operations in multiple "batches", but still it's manageable and not hard to implement
• Complex aggregations can be done with the standard aggregation pipeline or map-reduce

On the other hand, if you go with relational style, you'll be in a big trouble when you reinvent a SQL JOIN within your application code. This is one of the most common anti-patterns of using MongoDB (and such). Here's a very typical pseudocode:

for (HashTag tag: mongodb.hashtags.find()) {
   for (Image img: mongodb.images.find(
           new Document("_id", new tag.getImageId()))) {
       // ...
   }
}

This is inefficient, not scalable, and you are simply reinventing a wheel. Using this, you'll probably end up with complexity of O(N*M) because of loops within your code. If you'd choose SQL with foreign keys instead, you'd have something like O(N*log(M)) or even O(N+M).

There are no tables (relations) and foreign keys in MongoDB. Do not invent them, please. Use SQL instead, if you need. In fact, I highly suggest using SQL instead of MongoDB, unless your data really consists of documents.

Typical examples of documents are configurations, forms, and maybe user sessions. Those typically don't fit well into tables because of "random" structure.