That would be a part of feature selection. There are many methods to find out if there are relationships between the dependent variable and independent variables. To name a few: plots, measures of correlation, measures of mutual information.
The default quantiles with quantile are 0%, 25%, 50%, 75%, 100%.
This means that the quartiles Q1 and Q3 are the second and fourth value in the two vector you obtain.
Since indexes start at 1 it means that you can obtain Q1 with ar_quantiles and Q3 with ar_quantiles. Currently you're using indexes 1 and 2 and that's certainly what causes the problem.
There is an overall inverse relationship, but not a strictly monotone one. See e.g. the precision-recall curves in the sklearn examples.
A model that declares every record [to be positive class] has high recall but low precision.
If the model declares every record positive, then $TP=P$ and $FP=N$ (and $FN=TN=0$). So recall is 1; and the precision is $P/(...
Precision and recall are "hard" metrics. They are measure if the model's prediction is exactly the same as the target label.
Often times systems like yours can use a more flexible metric such as top-5 error rate, the model is considered to have generated the correct response if the target label is one of the model’s top 5 predictions.