Learn Hardcore French: Hier, Marie a dit qu'elle viendrait avec nous sur l'île. - Yesterday, Marie said that she would come with us to the island.

a dit is the passé composé, used for a single, completed action at a specific moment in the past.
disait is the imparfait, used for background actions, ongoing past situations, or repeated habits.
Here, Hier, Marie a dit... refers to one specific moment yesterday when she said this once, so a dit is the natural choice.
You would use disait in a sentence like Autrefois, Marie disait qu'elle viendrait, meaning she used to say that, repeatedly or habitually.

Why is viendrait in the conditional instead of a future like viendra or a present like vient?

The main verb a dit is in the past, and the coming is a future action relative to that past moment.
In French, this “future in the past” is normally expressed with the conditional: qu'elle viendrait = “that she would come.”
Using a simple future (qu'elle viendra) would present the coming as a future seen from now, rather than from the time when she spoke.
So a dit qu'elle viendrait keeps the time relationship clear: first she spoke, later (from her viewpoint then) she is supposed to come.

Is Hier, Marie a dit qu'elle viendra avec nous sur l'île ever possible, or is it always wrong?

It is not strictly wrong, but it is less standard.
qu'elle viendra here can sound colloquial and can suggest that, in the speaker’s mind, her coming is still a firm, relevant future plan at the moment of speaking.
Most teachers and grammar books recommend qu'elle viendrait after a past tense in reported speech.
So for careful or formal French, use qu'elle viendrait; in everyday spoken French, you may hear qu'elle viendra, but it is not the default in writing.

Why is it qu'elle and not que elle?

French avoids putting one vowel right after another at word boundaries.
So que becomes qu' before a vowel or a silent h: qu'elle, qu'il, qu'on, etc.
This shortening is called élision, and in these cases it is mandatory, both in writing and speaking.
Forms like que elle or que il would be considered incorrect; you must write and say qu'elle, qu'il, etc.

What does the little hat on l'île (the circumflex on î) mean, and does it change the pronunciation?

The circumflex in île (the î) mainly shows a historical change: there used to be an s after that vowel (compare English island).
In modern standard French, i and î are generally pronounced the same, so ile and île would sound identical.
However, the correct spelling is île with the circumflex.
In a few other cases (especially â, ê, ô), the circumflex can slightly affect vowel quality, but with î the effect is minimal or absent in most accents.

Why is it sur l'île and not à l'île or dans l'île?

Here l'île is just “the island” in a general sense, not a specific named island.
With this generic use, French naturally says sur l'île = “on the island.”
dans l'île is also possible but emphasizes being inside/within the island area or community, a bit like saying “within the island.”
à l'île is unusual when l'île is not followed by a name; à is more typical with names of places: à Tahiti, à l'île de Ré, à la Réunion, etc.
So in this neutral sentence, sur l'île is the normal preposition.

Shouldn’t dit agree with Marie and become dite, since Marie is feminine?

No. With the auxiliary avoir (as in a dit), the past participle normally does not agree with the subject.
It only agrees with a direct object if that object appears before the participle.
In this sentence, there is no direct object at all, so dit stays unchanged, even though Marie is feminine.
Example with agreement: Les choses qu'elle a dites – here dites agrees with les choses (feminine plural) placed before the verb.

Why is it qu'elle viendrait and not qu'elle vienne in the subjunctive?

Here a dit que introduces reported speech – it simply reports what Marie said.
For that, French uses the indicative (or, as here, the conditional as “future in the past”), not the subjunctive.
Marie a dit qu'elle viendrait = “Marie said that she would come,” just relaying information.
The subjunctive vienne would suggest wish, order, doubt, etc., and in that sense French normally uses dire de plus an infinitive: Marie a dit de venir (“Marie told [someone] to come”), not a dit qu'elle vienne in this context.

Can we move hier to another place in the sentence, or must it stay at the beginning?

You can move hier; several positions are correct:

Hier, Marie a dit qu'elle viendrait...
Marie a dit hier qu'elle viendrait...
Marie a dit qu'elle viendrait avec nous sur l'île hier.

The last version is grammatically fine but can be ambiguous, because hier might seem to modify viendrait (“she would come yesterday”).
To keep the meaning clear (that she said it yesterday), it’s better to put hier at the start or right after a dit.

How are qu'elle viendrait and sur l'île pronounced? Are there any tricky liaisons or silent letters?

Some key points:

Hier: [i.jɛʁ], roughly like ee-air.
Marie a dit: final t of dit is silent → [ma.ʁi a di].
qu'elle: said as one unit [kɛl] (kel); qu is just [k], not kw.
viendrait: [vjɛ̃.dʁɛ]; the final -t is silent, and ien gives a nasal vowel.
sur l'île: sur is [syʁ]; l'île is [lil]; together [syʁ lil] with no extra consonant added between sur and l'île.