What does "може" mean
1) What does "може" mean in the following sentence?

И аз може да стана като възрастните, които не се интересуват от нищо друго, освен от цифри.

As I know, "може" is used with third person subject.
аз should be used with "мога".
Can "може" be used instead of "мога"?

2) What does "си" mean in the following sentence ?

Ето затова си купих кутия с бои и моливи.

Question

What does "може" mean
1) What does "може" mean in the following sentence?

И аз може да стана като възрастните, които не се интересуват от нищо друго, освен от цифри.

As I know, "може"  is used with third person subject.
аз should be used with "мога". 
Can "може" be used instead of "мога"?

2) What does "си" mean in the following sentence ?

Ето затова си купих кутия с бои и моливи.

Answer

1) Yes, it can. Of course, in that sense, it isn't 1st. peson conjugation, but speaking in general, without a subject (we use 3rd p.s. in such cases), so the meaning is "it is possible" and not "I can" - most of the time they are interchangeable, though.

2) It's a reflexive particle, (showing that the verb action is directed to its subject), in this case " I bought a box for myself"