Numerical checks are more reliable than visual checks.
There is no easy answer to that. The first step is to diagnose mixing, and then to check that you have enough many iterations to report the desired number of digits. There is no magical threshold that would guarantee good mixing, so it’s always some compromise. See How many iterations to run and how many digits to report, for how to check whether the number of posterior draws is big enough for the desired number of digits to report.
In automated approach, \hat{R} has the downside that you are looking at many quantities which have some variability given finite sample size and thus more quantities you have more likely it is you see false positives, and then you would need to either adjust the sample size or the threshold.
R^* has the benefit of providing just one value while examining the joint posterior. If R^* looks good, then focus on MCSE as discussed in How many iterations to run and how many digits to report.