Learn Spanish (Latin America): Me encantaría que el museo exhibiera la colección para examinarla. - I would love it if the museum exhibited the collection to examine it.

The main verb encantaría is in the conditional tense (I would love). When expressing a wish or emotion using the conditional, the verb in the que clause is forced into the imperfect (past) subjunctive to match. If the main verb were in the present tense (me encanta que...), you would use the present subjunctive exhiba.

Could I use exhibiese instead of exhibiera here?

Yes! Both are perfectly valid forms of the imperfect subjunctive. However, the -ra ending (exhibiera) is overwhelmingly more common in spoken Latin American Spanish, making it the best one to focus on.

Why is examinarla in the infinitive instead of using another subjunctive phrase like para que la examine?

Because the person doing the examining is the same person expressing the wish. When there is no change in subject, you use the preposition para followed directly by an infinitive verb (examinar). You only use para que plus the subjunctive when a new subject is introduced (for example, so that the students could examine it).

What does the -la at the end of examinarla refer to?