I am not a native speaker, but I think it is because the name is under the locative case declension. "na Katerinu" yes means "about Katerinu" in this context, but it literally means "on Katerinu" (at least that's how I understand it from an English > Czech perspective). In which case, "on Katerinu" means that Katerina is not the nominative case in this sentence. We are the nominative case because we are thinking about her. Thus, her name must be declined. You will see this throughout Czech with all names. They even have a vocative case, which means the name must be changed when you are speaking directly to someone (for example, when somebody is talking about me they say Max, but when somebody is talking to me they say Maxe). Hope this helped
Yes, adding the prefix po- is one of the ways how you can make perfective verb (myslet → pomyslet). However, there is a difference between "pomyslet" and "pomýšlet"; "pomýšlet" is imperfective... sometimes the rules are just illogical, I know.
Other verbs, that use prefix po- to create perfective pair, are for example:
bavit → pobavit (to entertain)
chválit → pochválit (to compliment)
čekat → počkat (to wait) (e gets dropped)
We discussed grammatical aspect of verbs. From wikipedia:
"Aspect is a grammatical category that expresses how an action, event, or state, denoted by a verb, extends over time."
https://en.wikipedia.org/wiki/Grammatical_aspect (check it for clearer and more in depth explanation)
To make it easier to understand, there are some similarities between Czech grammatical aspects and English continuous/simple tense (not the same tho).
Myslel jsem na ni. → I was thinking about her. (imperfective)
Pomyslel jsem na ni. → I thought about her. (perfective)
But this rule doesn't always work. Czech aspects are more chaotic and messy, and a lot of the times by making a perfective verb an imperfective you change the meaning significantly. Some verbs don't even have their imperfective or perfective pair.
Don't worry about it too much though. In time this course will slowly teach you how to use and conjugate verbs properly :-).
Usually, perfective aspect of a verb makes an action "finished", while imperfective does not necessarily specify if it has been finished or not. Giving you a Polish example here - "sprzątał" means "he was cleaning" - we don't know if he had cleaned all the room or only a part of it; "posprzątał" means "he cleaned" - here we are sure that he finished the action and the room is already clean.
I think the use of se is lexical, that is you have to learn it with the verb. Myslít (not sure about the infinitive form) is simply not one of those verbs that are constructed with se. The "se" is probably a reflexive pronoun which is used in this form irrespective (I resisted the temptation to write irregardless there :) ) of the person. In other languages the pronoun is different according to the person, for example, me, te. etc.