How to choose our data set wisely?

You have got nothing to lose by framing it as a multi-label detection task. All you have to do is remove the final Softmax layer and format your ground truth as multi-hot, and you can regress multiple labels directly. Mean Squared Error and Sum of Absolute Differences are neutrally biased with respect to false positives and false negatives if the data are balanced, whereas categorical cross-entropy is biased for higher recall and lower precision and usually does not work as well once the Softmax is removed, so choose your loss function accordingly.

Definitely include more objects in your training set that you don't want to detect--this will reduce overfitting by forcing the detection template(s) to improve their specificity.

I would think of this as a object detection problem with three (3) classes. I would initially train the model without images that don't have any of the target classes (no bounding boxes). Rather I would include images that have at least one of the target classes (with bounding boxes) and few of the non-target objects. Eventually, I will test the model with unseen images including those images that don't have any of the target classes. If the model doesn't perform well, I would try to address the reason and retrain accordingly (most likely training again with more images per class, balanced dataset etc.).

Also, I think the model would predict the non-target objects as one of the target classes with low confidence and likewise it will predict the correct objects with higher confidence. If that's the case, I would increase the confidence threshold accordingly which will restrict the model from detecting low-confidence non-target objects.