疲れる is not an unusual verbal, so there is no need to identify verbals like it. Rather, understanding how Japanese expressions tend to be expressions of changes of state will help with interpreting this and similar future encounters with perfective ("past") forms.
1) Why is 疲れた used to indicate a present state? (Is there a logic behind these types of verbs or is it easier to remember these verbs?)
た expresses completion. We use this regardless of the tense. As @psosuna says, 疲れた means I have become tired, and this form of English is called the present perfect; the tense that this form of the perfect aspect has is also the present. Both are to talk about a past event as a present state/fact/condition. (The 'become' after 'have' is the past participle, which is a non-finite verb.)
2) What is the difference between 疲れたand 疲れている? Are there situations where one would be preferred over the other?
We say 疲れた at the moment we feel like wanting to utter this word, in other words when we have just become tired (i.e. a change of state has occurred), in reality though we would keep saying 疲れた for any numbers of days. We say 疲れています when we've been feeling tired for a while, ...maybe we start saying this after getting tired of keeping saying 疲れた.
Remember that forms in -ている refer to an activity or state that something is/will be in, which results from a completed occurrence, and which extends over a period of time: e.g. 疲れた (becoming tired) already occurred, and my resulting state (being tired) extends over a period of time when I want to say
3) What would 疲れる mean? Would it be "I will be tired" or "I'm getting tired"?
Yes. We say 疲れるからやめて, and here we mean "Please stop it because I'll be tired." "疲れる" is often used as complaint, meaning it makes me tired or it's making me tired.
4) Japanese also uses time-related words like 毎朝 or 今. Does using these words change the state choice as well?
I expect so, since Japanese language doesn't differentiate tense by the verb forms, these adverbs of time or other additional expressions are very important. But it seems we understand each other more by the context in everyday life situations. For example, no one say 今、疲れちゃった, but we understand you mean now from the context without hearing 今.
For example: I feel like it should be 今、疲れている。 And 毎朝、疲れた。when using different time-related words.
What I can come up with right now are 毎朝、疲れを感じます or 最近、なんだか疲れやすいです or 今、ちょっと疲れてるから、あとでね.