Thanks for your questions! Just a note: when you have questions on more than one topic, it may be helpful to create a new thread for each topic, with a descriptive title - that way, when others in the forum have the same question, they will be able to see that someone else has already asked it.
Now, let's get to your questions!
1. Remember what we were saying in an earlier thread: using an automatic translation tool when you're first learning a language can be a tricky road to go down, because you will often get general translations - that means that you will end up with several phrases that translate to the exact same thing, or even translations that skip over words. If you do want to use an automatic translation tool and you find that you are getting identical translations for different sentences, the best thing to do is to look at the words that are different in your two sentences.
Here, the parts that are different between these two sentences are the words abmachen and vereinbaren. If you look these up in a dictionary or other language site, you will find that abmachen can translate to "to arrange / to agree," and vereinbaren can translate to "to agree / to arrange / to schedule." So, as you can see, they are very similar in meaning, and so there is no real difference between Ich würde gerne einen Termin abmachen and Ich würde gerne einen Termin vereinbaren - it's just a matter of vocabulary preference.
2. The difference between mich and mir is that mich is in the accusative case and mir is in the dative case. The issue is the same with dich and dir: the first is accusative, and the second is dative. If you need a refresher on the differences between the accusative case and the dative case, you can check out the lessons in Level 1, Module 6 on each of these cases, where you will find a good description of how each case is used and what their differences in meaning are. If you haven't got that far in the course yet: don't worry! Get used to using them in set phrases, and we'll get into the details of how they work further along in the course.
I hope that these answers have been helpful!