First, I think regarding the mincor argument in quickpred, it will decide to only use variables as predictors in the imputation model, that are correlated at least r=0.1 with the target variable (note that mincor = 0.1 is the default). As such, if you use the argument as mincor = 0, it should use all variables as predictors in the imputation model (assuming there are no errors - as I have never tried this).
Second, you would have to first diagnose the missing data mechanism (i.e., is it missing completely at random - MCAR, missing at random - MAR, or missing not at random - MNAR).
- MCAR: The probability of missing is the same for all cases (i.e., causes of the missing data are unrelated to the data).
- MAR: The probability of being missing is the same only within groups defined by the observed data.
- MNAR (or NMAR): Missing not at random or not missing at random means that the probability of being missing varies for reasons that are unknown to us.
Generally speaking, MCAR is usually unrealistic, MAR is somewhat plausible, while MNAR is often plausible. As such, MCAR and MAR are typically considered "ignorable" as information about the missing data itself is not included when dealing with the missing data. In contrast, MNAR is typically "non-ignorable" because the missing data mechanism must be modelled while you deal with the missing data.
Once you have diagnosed the missing data mechanism it will inform your approach to dealing with the missing data. For example, using multiple imputations (MI) and maximum likelihood (ML), you assume your data is at least MAR. Listwise deletion requires data to be MCAR to reduce/prevent bias in results. If you aim to use MI, you assume your data is at least MAR. If your data is MNAR, then using MI is not appropriate.
Diagnosing the mechanism: https://www.theanalysisfactor.com/missing-data-mechanism
Regarding how much data MI can "deal with," the MICE package should be able to deal with the missingness you have mentioned above. However, you should note that the more missingness for a particular variable will be reflected in larger error terms when compared to those variables with fewer missing data points. This will impact your ability to detect significant relations to these variables.