1. Differences between あそこ VS むうこ VS あっち/あちら

for asoko (and koko/soko/doko) it is referring specifically to a place, while achira is more along the lines of "that way" or "that person" or "that side"
mukou is for the opposite side/place usually


2. Differences between どなた VS だれ

donata is more polite

3. Differences between TO HAVE (ある　VS もっている)

usage depends mostly on what the subject of the sentence is. if the item that is had is the subject, you can say (item) ga aru, but if the person who has the item is the subject, you say (person) wa (item) o motteiru

4. What would you translate this as 子供の時、　いいこどもでしたか。(Were you a good child? -my translation)
   Also how would you answer this?

that's right
はい、もちろんだよう。

5. Difference between 山下先生はあそこです。VS　あそこに山下先生があります。

the first one focuses more on Yamashita- "as for Yamashita Sensei, he is over there" while the latter is more of just stating a dry fact

6. いつクラスがありますか。(When do you have class? - my translation)
　 Is いつ used for a date/day rather than a time?

you can use itsu for date/day and time, it's just like the English "when". But nanjikan is more specific if you want the time
   What would be a response?
三時半にあります。　<- it's at 3:30.


7. 次の人はわたしといっしょにきてください。(Next person please come with me. -my translation)
   What would you translate this as or is there a better way to phrase it?

i would probably leave off isshoni. also a phrase like that would usually be said in super honorific language (which sucks and is very stupid don't worry about it til later)

8. ゆっくりはなしてください。（Please speak slowly. -my translation)
   Would this work and also where would ゆっくり go in the sentence if there was a direct object?

could go before the D.O. or stay before the verb

ゆっくり日本語を話してください
or
日本語をゆっくり話してください

IMO option two sounds better but it works either way