Learn Spanish (Spain): Dudo que ella entienda el algoritmo. - I doubt that she understands the algorithm.

The verb dudar expresses uncertainty. When a main clause expresses doubt about an action in a second clause introduced by que, that second verb must take the subjunctive mood.

Would we still use the subjunctive if the sentence started with No dudo que?

No, you would use the indicative mood and say No dudo que ella entiende el algoritmo. When you make dudar negative, it expresses certainty or belief, which triggers the indicative rather than the subjunctive.

Is it mandatory to include the pronoun ella in this sentence?

It is highly recommended for clarity here. In the present subjunctive, the conjugated forms for yo, él, ella, and usted are identical. Without ella, the sentence Dudo que entienda el algoritmo could easily be misunderstood as expressing doubt about your own understanding.

Why does entienda have an ie in the middle when the infinitive is entender?

The verb entender is a stem-changing verb where the stress causes the e to become ie. This exact same vowel change carries over from the present indicative to the present subjunctive for all subjects except nosotros and vosotros.