First I answer your first question while thinking the rest. You have $\max\{\overline{X},1\}$ as an estimator. That is, you find the sample mean of your data which is supposed to be $\overline{X}$ and take the maximum of $\overline{X}$ and $1$.
This is called your estimator. I havent checked if it is a maximum likelihood estimator yet. If so, it should minimize the likelihood function.
Now you have an estimator (probabily maximum likelihood) which is trying to estimate the maximum of the $\mu$, the parameter of your distribution and $1$.
EDIT: Okay here is the idea: You want to estimate the maximum of $\mu$ and $1$. There might be two cases. Either $\mu>1$ or $\mu<1$. In the first case $\max\{\mu,1\}=1$ else is $\mu$.
Assume that $\mu<1$ then we have $\max\{\mu,1\}=1$. This means our objective is to estimate $1$ from given samples. Now lets have a look at our estimator. This estimator will give you less than half of the times the value that you have and else you will get $1$. When you check the expected value then it will be something less than $1$ which indicates a Bias towards the positive real axis.
On the other hand when $\mu>1$, we have $\max\{\mu,1\}=\mu$. But we have real problem! our estimator will output more than half of the time a number which is greater than $1$ and less than half of the time a value which is less than $1$ which will be eventually rounded to $1$ due to its structure $\max\{\mu,1\}$. As you can see, we want to estimate $\mu$ and unfortunately all the values which are negative and between $\mu$ and $1$ are rounded to $1$. That is not good! this will result a Bias towards the positive real axis. Because in order not to have a Bias, we need to have equally from the values that are less than $\mu$ and greater than $\mu$ in the order. Not the time of occurance. As all the negative values will be mapped to $1$, this estimator will never converge to $\mu$ but something greater than $\mu$.
To show this theoretically. You need to determine the expected value of this estimator in both cases and show that it deviates from the true (to be estimated) value.