Describe a TM through denotation of the transition function
You are making way too many guesses, and seem to be confused on the difference between words (elements of $\Sigma^*$, or $\Gamma^*$) and letters (elements of $\Sigma$ or $\Gamma$). This is basic stuff and need to be understood before even pretending to think about transition functions.

Indeed, the Turing machine gets a string on its input tape (with letters from $\Sigma$) flanked by $\Box$'s. To be precise $w$ is not an element of the tape alphabet, but a string over the input alphabet $\Sigma$. The tape alphabet equals $\Gamma = \{ 0,1,\#,\Box \}$ plus all the additional symbols that you need for the task you have to perform.

The task is: check whether the input is in fact a string of the form $w\#w$, i.e., two copies of the same string separated with $\#$. The Turing machine may write on the tape and use a little bit of memory, but cannot read stings $w$ in one go (which is what your question suggests). It makes steps on single letters like $(p,0,p_0,\bar 0,R)$: "in state $p$ I read letter $0$, switch to state $p_0$ [to remember $0$] write $\bar 0$ on the tape [replacing $0$ to indicate I have seen it] and move to the position on the right". (Now you have to add $\bar 0$ to $\Gamma$.)

So check your lecture notes on the technical details again, and start thinking how to do the comparison letter-by-letter. It is not that hard: just pretend you are the machine and analyse your steps on a simple example input.