Does associativity imply commutativity? I used to think that commutativity and associativity are two distinct properties. But recently, I started thinking of something which has troubled this idea:
$$(1+1)+1 = 1+ (1+1)\implies 2+1=1+2$$ Here using associativity of addition operation, we've shown commutativity. 
In general, $$\underbrace{(1+1+\dots+1)}_{a \, 1\text{'s }}\ \ + \ \ \underbrace{(1+1+\dots+1)}_{b \, 1\text{'s }}=\underbrace{(1+1+\dots+1)}_{b \, 1\text{'s }}\ \ + \ \ \underbrace{(1+1+\dots+1)}_{a \, 1\text{'s }} \\ \implies a+b=b+a$$
For any natural $a,b$.
Hence using only associativity we prove commutativity. That this can be done, is disturbing me too much. Is this really correct? If yes, then are associativity and commutativity closely related? Or is it because of some other property of natural numbers? If yes, then can it be done for other structures as well?


using only associativity we prove commutativity

Well, not really. You also used your knowledge that any natural number (but not zero or negative numbers!) can be expressed as a sum of 1s. And you didn't prove "commutativity" in general; you proved "commutativity of addition (of natural numbers)", which you already knew to be true anyway.
A simple example of an operation that is associative but not commutative is "addition of strings" as defined in most programming languages — that is, string concatenation. Concatenation is associative:
("abc" + "def") + "ghi" = "abcdefghi"
"abc" + ("def" + "ghi") = "abcdefghi"

However, obviously, it's not commutative:
"abc" + "def" = "abcdef"
"def" + "abc" = "defabc"

And vice versa: an example of an operation that is commutative but not associative is $x\star y = xy+1$.
$
   (1 \star 2) = 3 \\
   (2 \star 1) = 3 \\
\text{but} \\
   (1 \star 2) \star 3 = 3 \star 3 = 10 \\
   1 \star (2 \star 3) = 1 \star 7 = 8
$
