1. Mother told me, "Not to take a long nap."

2. Mother told me,"Don't take a long nap."

(I think #2 is right? What about #1?)

3. Every time I have to read out loud, I get so nervous.

4. Each time I have to read out loud, I get so nervous.

5. Whenever I have to read out loud, I get so nervous.

6. When I have to read out loud, I always get so nervous.

(Are the four sentences all the same?)

7. Please be on time.
8. Please you be on time.

(#7 is right? How about #2? Can we say like that as in #8?)

9. How many a's does the word have?

10. How many as does the word have?

11. How many 'a's' does the word have?

(Among #9 and #11, is #9 only right? Can we have another usage for writing the plural form of the alphaber a?)

1. Mother told me not to take a long nap.

2. Mother told me, "Don't take a long nap."

(I think #2 is right? What about #1?)
I corrected #1. And yes, #2 is correct.

3. Every time I have to read out loud, I get so nervous.

4. Each time I have to read out loud, I get so nervous.

5. Whenever I have to read out loud, I get so nervous.

6. When I have to read out loud, I always get so nervous.

(Are the four sentences all the same?)
Yes, all four are the same. One thing, though -- when you use "so" like this, the sentence should be followed by a result clause: Whenever I have to read out loud, I always get so nervous that my hands sweat.

7. Please be on time.
8. Please you be on time.

(#7 is right? How about #2? Can we say like that as in #8?)
#7 is correct, but #8 is not. The "you" needs to be silent (also referred to as "understood").

9. How many a's does the word have?

10. How many as does the word have?

11. How many 'a's' does the word have?

(Among #9 and #11, is #9 only right? Can we have another usage for writing the plural form of the alphaber a?)
Only #9 is correct.
http://grammar.ccc.commnet.edu/grammar/marks/apostrophe.htm
Scroll almost all the way down to read about using apostrophes to form plurals. It's used sparingly for this purpose!