Hi, David.
1. She is as pretty as other/ the other girls you've met before. -- both OK (the is OK if the speaker presumes the listener has met girls previously)
He easily gets distracted by other/ the other students. -- both OK (the is OK if the context is a class they are already talking about).
Some girls are beautiful, others are just plain-looking. -- No the.
2. Please check the box at /on the left on the computer. -- Both OK. Of the computer?
3. When do you use a singular noun after any? Could you please give me some examples? -- Usually, some stress is required: 'I don't have any friend at all in Toledo'.
4. Do you put a plural noun or a singular noun after any other? -- Same guideline as for #3 above.
5. Are the following sentences correct?
1. I invited him to come over. -- OK
2. A: How are you gonna get here? -- OK
B: I will drive a car over -- OK C:I will take a bus over. -- OK
3. He is going to move over/ over there. -- OK
4. Do you want me to walk you over? -- OK
Could you please explain to me how to use over here? -- There is away from the speaker; here is near the speaker. 'Come over here so that I can hit you'.
I'm also a little confused with when to use other or the other.. -- Use the when there is a specific or previously mentioned other; the rule is no different from that for the alone.