Dubito che il supermercato sia ancora aperto.

Because dubito che normally triggers the subjunctive in Italian.

• è = present indicative of essere
• sia = present subjunctive of essere

After expressions of doubt, uncertainty, emotion, opinion, or possibility, Italian often uses the subjunctive. Since dubito che means I doubt that..., the clause after che takes the subjunctive:

• Dubito che il supermercato sia ancora aperto.

Using è here would sound non-standard in careful Italian.

Sia is the present subjunctive of essere.

Here is the full present subjunctive of essere:

• che io sia
• che tu sia
• che lui/lei sia
• che noi siamo
• che voi siate
• che loro siano

In this sentence, il supermercato is third person singular, so Italian uses sia.

Che introduces the subordinate clause: that the supermarket is still open.

So the sentence is structured like this:

• Dubito = I doubt
• che il supermercato sia ancora aperto = that the supermarket is still open

This is very common in Italian:

• Penso che...
• Credo che...
• Spero che...
• Dubito che...

Because aperto agrees with il supermercato, which is masculine singular.

• il supermercato = masculine singular
• therefore: aperto

If the noun were feminine singular, you would use aperta:

• Dubito che la farmacia sia ancora aperta.

If it were masculine plural:

• Dubito che i negozi siano ancora aperti.

Here ancora means still.

So:

• sia ancora aperto = is still open

In other contexts, ancora can also mean again:

• Puoi dirlo ancora? = Can you say it again?

In this sentence, though, the meaning is clearly still.

In standard Italian, no — not if you want to speak and write correctly.

After dubito che, the expected form is the subjunctive:

• Dubito che il supermercato sia ancora aperto.

You may sometimes hear the indicative after words that normally take the subjunctive, especially in informal speech, but learners should treat that as something to recognize, not imitate.

Because Italian usually uses the definite article more often than English does.

• il supermercato = the supermarket

This can refer to:

• a specific supermarket already known from context, or
• the supermarket that is relevant in the situation

Italian often prefers the article where English might sometimes omit it in a looser phrase.

Yes. Dubitare can be followed by different structures.

1. Dubitare che + clause

   • Dubito che venga.
   • I doubt that he/she is coming.

2. Dubitare di + noun / pronoun / infinitive

   • Dubito di lui. = I doubt him.
   • Dubito di poter venire. = I doubt I’ll be able to come.

In your sentence, there is a full clause with its own subject (il supermercato), so che is the natural choice.

Dubito che... is perfectly correct and natural. It sounds a bit clear and deliberate, like I doubt that...

Depending on context, Italians might also say:

• Non credo che il supermercato sia ancora aperto.
  = I don’t think the supermarket is still open.

• Secondo me il supermercato non è più aperto.
  = In my opinion, the supermarket isn’t open anymore.

• Mi sa che il supermercato è chiuso.
  = I think the supermarket is closed.
  (more informal)

So dubito che is natural, but not the only option.

It is historically the past participle of aprire, but in this sentence it functions very much like an adjective meaning open.

• il supermercato è aperto = the supermarket is open
• il supermercato è chiuso = the supermarket is closed

This is a very common pattern in Italian. Learners often first meet aperto as part of compound tenses, but here it describes the state of the supermarket.

A little, yes, but the original order is the most neutral and natural:

• Dubito che il supermercato sia ancora aperto.

You might also hear:

• Dubito che il supermercato sia aperto ancora.

But this is generally less neutral, and ancora aperto is the more standard and natural grouping.

So for learners, the safest version is:

• sia ancora aperto

Yes. The subjunctive form of essere has to agree with the subject.

Examples:

• Dubito che noi siamo pronti.
  = I doubt that we are ready.

• Dubito che loro siano a casa.
  = I doubt that they are at home.

In your sentence, the subject is il supermercato, so the correct form is sia.

Word for word, it is roughly:

• Dubito = I doubt
• che = that
• il supermercato = the supermarket
• sia = be / is (subjunctive)
• ancora = still
• aperto = open

So the structure is:

• I doubt + that + the supermarket + is still open

That is why the sentence feels quite close to English in meaning, even though the grammar differs because Italian uses the subjunctive after dubito che.