Is it decidable "Given a TM M, whether M ever writes a non blank symbol when started on the empty tape."
Q1: A state can get repeated. The point is that if a state gets repeated and no non-empty symbol has even been written, then you known the that the Turing machine will never halt as it is necessarily stuck cycling through some of the states encountered so far. Since none of the states of the cycle caused the TM to write a non-blank symbol, the TM will never write a non-blank symbol.

After $2q$ steps, either a non-blank symbol has been written, and you can answer "yes", or all the written symbols were blank and (at least) one state must have been encountered two or more times, meaning that you can answer "no".

Q2: My guess is that $w$ is defined somewhere to be the empty string.
If the TM can be started with some arbitrary string $w$, then a variation of the above solution still works.
Suppose that the head of the TM starts at the beginning of $w$.
The number of states increases by a factor of at most $1+|w|$ in order to take into account all possible states of the tape (if only blank symbols are written, then the tape always contains one of the $1+|w|$ suffixes of $w$). If the head can start anywhere then this factor will be at most $1+|w|^2$.