String descriptions of machines
The exact description of how he converts an instruction to a string is:
In this description we shall replace $q_i$ by the letter "D" followed by the letter "A" repeated $i$ times, and $S_j$ by "D" followed by "C" repeated $j$ times.
So the instruction $q_1 S_0 S_2 L q_2$, representing "If we are in state $q_1$ and the current space on the tape is blank, write a 1, move one space to the left and go to state $q_2$", would be written as "DADDCCLDAA".
If by "print nothing" you mean "print a blank", then that is represented by the symbol "D" followed by zero "C"s. However, if by "print nothing" you mean "leave the current symbol unchanged", it has no explicit representation in Turing's notation but instead you would have an instruction that includes $S_i S_i$, i.e. "if the current symbol is $S_i$, then erase it and print the symbol $S_i$". If you have an entire state that leaves the current space unchanged, then that would involve having one of these instructions for each possible symbol in this particular state.
The decision machine
In saying that $\mathcal{H}$ has its motion divided into sections, Turing is saying that it performs an iterative process that operates over the numbers $1, 2, 3, \ldots$. He is defining a function $R$ that returns "the number of circle-free machines represented by the integers from $1$ to $N$". It does so by simply writing down the integers, and using the decision machine $\mathcal{D}$ to check whether the most recently written integer is the DN of a circle-free machine. If it is, then $R(N) = R(N-1) + 1$, and if it isn't then $R(N) = R(N-1)$. There is an implicit $R(0) = 0$ to start the process off.