My understanding is that if the verb "espérer" is negated like "Je n’espère pas qu’il cherche …", then this "cherche" should be the subjunctive.

No, it doesn’t have to be the subjonctive.

But how about when the verb "chercher" is negated instead, as shown above? Is this the Indicatif Présent?

In your example it’s indicatif present, yes.

In the following examples I doubled the verb chercher with suivre since chercher has the same conjugation in indicatif présent and subjonctif présent (except for nous and vous).

J’espère qu’il cherche/suit son frère = I hope he is looking for/following his brother

J’espère qu’il ne cherche/suit pas son frère = I hope he isn’t looking for/following his brother

Je n’espère pas qu’il cherche/suit son frère = I don’t hope he is looking for/following his brother

Je n’espère pas qu’il ne cherche/suit pas = I don’t hope he isn’t looking for/following his brother

They are all at present tense. There is the idea of right now, it’s happening.

Now I know what you mean, it can be the subjunctive, there is something about the subjonctive with the verbs vouloir, souhaiter, espérer, etc. I’ll let this answer to more qualified people.