Elle pardonne à Paul vs. Elle lui pardonne

The answer directly to your first question is 'yes' both are correct - the second sentence would be used when it is already known to the reader/listener, through context, that 'lui' is Paul. 

By definition, an object (the grammatical word 'object') that follows a preposition (à/pour) after a verb, is an indirect object, so you can never use "à + direct object". 

For an indirect object pronoun replacing "à + noun", the indirect object pronoun is used in front of the verb. (Of course, a direct object pronoun also precedes the verb in French.)

In terms of when does a verb take a direct object or indirect object - unfortunately, that just takes time to become familiar with the minority of verbs that take an indirect object - there are lists you can look at to get a head start, as noted in the 3rd link below, and a good way is to acquire them naturally as you come across them.

You may benefit from also reading the first 2 articles on object pronouns. 

https://kwiziq.learnfrenchwithalexa.com/my-languages/french/glossary/2   

  https://kwiziq.learnfrenchwithalexa.com/my-languages/french/glossary/2 

https://www.cliffsnotes.com/study-guides/french/french-i/french-i-prepositions/verbs-requiring-indirect-objects