Learn Hardcore Spanish (Spain): Espero que mi salario llegue a tiempo.

Because esperar que expresses a hope/wish, which triggers the present subjunctive. So you need llegue, not the indicative llega. Rough guide: verbs of desire, emotion, doubt, recommendation, etc., take the subjunctive in the dependent clause.

Can I drop the que and say: Espero mi salario llegue a tiempo?

No. After verbs like esperar when you introduce a clause, Spanish requires the conjunction que. Without it, the sentence is ungrammatical in standard Spanish.

Is mi salario what people in Spain normally say? What about sueldo, nómina, or paga?

sueldo: the most common everyday word in Spain for “salary/pay.”
salario: correct, slightly more formal/technical.
nómina: literally the payslip/payroll, but often used metonymically for the monthly pay (especially when talking about bank deposits: ingresar la nómina).
paga: pay; also used for the extra payments (paga extra).

So you’ll often hear: mi sueldo, or in banking contexts: la nómina.

Should I add an indirect object pronoun and say Espero que me llegue el salario a tiempo?

You can. Me marks that the salary “arrives to me,” which is natural if you’re thinking of a bank deposit. All are acceptable:

Espero que mi salario llegue a tiempo.
Espero que me llegue el salario a tiempo. The second sounds a bit more idiomatic in everyday talk.

Are there even more natural ways to express this in Spain?

Common alternatives:

Espero que me paguen a tiempo. (They pay me on time.)
Espero cobrar a tiempo. (I hope to get paid on time; same subject, so infinitive.)
Espero que me ingresen la nómina a tiempo. (Very natural for bank transfers.) All are slightly more idiomatic than talking about the salary “arriving.”

Can I change the word order to Espero que llegue mi salario a tiempo?

Yes. Both …que mi salario llegue… and …que llegue mi salario… are correct. Putting the subject after the verb (llegue mi salario) can add a touch of emphasis or sound a bit more formal/literary, but it’s fine in speech too.

What’s the difference between esperar que and esperar a que?

esperar que + subj. = to hope that: Espero que todo salga bien.
esperar a que + subj. = to wait until: Espero a que llegue el salario. (I wait until the salary arrives.)
Also, without a clause, esperar can mean “to wait for” a noun: Espero el salario.

Does esperar que ever mean “I expect that…”?

In everyday Spain Spanish, espero que is usually “I hope that…”. For “I expect that…”, prefer:

Supongo que mi salario llegará a tiempo.
Imagino que… / Me imagino que…
Es de esperar que… (set phrase, somewhat formal) Note: Espero que mi salario llega a tiempo (indicative) is not idiomatic.

Is a tiempo the best way to say “on time”? Could I use something like en tiempo or puntual?

Use a tiempo for “on time.” Alternatives:

puntualmente (adverb): que me paguen puntualmente.
a su hora / a su debido tiempo have different nuances (“at the proper time,” “in due course”).
Avoid en tiempo here.

Why is llegue spelled with gue? And what’s the difference between llegue and llegué?

The u in gue is silent; it preserves the hard g sound before e.
llegue (no accent) = present subjunctive (or formal command).
llegué (accent) = first-person preterite (“I arrived”). Don’t confuse them.

How do I pronounce the tricky parts?

ll in Spain is typically pronounced like English y: [ʝ] (yeísmo), so llegue ≈ “yeh-geh.”
gue has a hard g; the u is silent.
ti in tiempo forms a diphthong: “TYEM-po.”

How does this change in the past?

Use past + imperfect subjunctive:

Esperaba que mi salario llegara/llegase a tiempo.
If you mean the arrival happened before the moment of hoping (rare here), use the perfect subjunctive:
Espero que haya llegado a tiempo.

When should I use the infinitive instead of que + subjunctive?

When both clauses share the same subject:

Espero llegar a tiempo. (I hope I arrive on time.) But if the subject changes (here it’s “my salary”), you need que + subjunctive:
Espero que mi salario llegue a tiempo.

Is de que ever correct here? What about Espero de que…?

No. With esperar, saying de que is a classic error (dequeísmo). Say Espero que…

Could I say para que instead of que?

Not here. Para que expresses purpose:

Trabajo duro para que me paguen a tiempo. (I work hard so that they pay me on time.)
Your sentence expresses a hope, so (espero) que is the right connector.

Any colloquial ways to say this in Spain?

Yes:

A ver si me pagan a tiempo.
A ver si me llega el sueldo a tiempo. These sound natural and slightly less formal.