How can I prove that every finite product can be transformed to the given form? Suppose, the permutations $a=(123)$ , $b=(12)(34)$ , $c=(12345)$ and $d=(12)(35)$ are given. I checked with GAP that the elemts 
$$a^jb^kc^ld^m$$ with $0\le j\le 2$ , $0\le k\le 1$ , $0\le l\le 4$ , $0\le m \le 1$ form the alternating group $A_5$.
Futhhermore, I found the following equations for $a,b,c,d$.
$$ba=ac^3d$$
$$ca=abc^2$$
$$da=bc$$
$$cb=a^2bc^2$$
$$db=ac^4$$
$$dc=c^4d$$

I want to prove BY HAND that with these equations, every finite product of the elements $a,b,c,d$ can be transformed to the form given above, of course, without using that the products above form a group.

It is sufficient to prove that any product of the form $a^jb^kc^ld^m$, multiplied with $a,b,c$ or $d$ can be transformed to such a product again.
The multiplication with $d$ is harmless. 
The multiplication with $c$ is also easy to handle because, if we have a product $dc$ at the end, it can be transformed into $c^4d$
If we multiply a product with $a$ or $b$, the situation is far less obvious. 
 A: I leave this for reference, but it is not a full answer.
=====
Consider the following process:
Take an arbitrary word in $a,b,c,d$. Whenever letters are in the wrong order (e.g. a $c$ in front of an $a$, use the left hand side of one of your equations to replace it with the right hand side. Repeat until the symbols are in the correct arrangement.
If you can show that this process will always terminate you are done -- you simply use the element orders to reduce exponents.
The standard method to show termination of such rewriting rules is to use ecial ordering on words (called a wreath product ordering) as below. Then whenever one of your identities is applied the word gets smaller in this ordering, furthermore one can show (I omit this proof here) that there cannot be an infinite chain of smaller and smaller words with respect to this ordering. This proves that the process terminates.
Alas, (as Derek Holt pointed out) this alone will not work with relations that make a word nominally larger (such as $cb=a^2bc^2), so for this uno extra treatment is needed.
==========
For the definition of the ordering we just need to describe how to compare two words:
First replace any block of symbols that does not involve $a$ in both words by a new symbol $z$, e.g. $bc^4a^3cd^5bab$ becomes $za^3zaz$. Compare the results of the substitution lexicographically. If they are different this is the ordering.
Otherwise the $a$-bits in both words are the same. Now you take the words and replace any block of letters $c,d$ by $z$ and again compare. If the substituted parts are the same, both $a$ and $b$ part of the words are the same.
Repeat the same for $d$ (i.e. both $d^{17}$ and $d$ would be replaced by a single $z$.) Compare. 
If $a,b,c$-parts are the same compare the full words lexicographically.
(A more formal definition and missing roofs can be found at
https://books.google.com/books?id=k6joymrqQqMC&lpg=PA138&ots=1-H556FtLm&dq=wreath%20product%20ordering%20automata&pg=PA133#v=onepage&q&f=false
)
