Noticing, observing and watching are greatly facilitated by an alert mind. If eyes fall then notice this as bodily phenomena. When you notice the falling of the eyes, what thoughts accompany that action? Watch carefully. You may find that mind is making something out of nothing or that it’s wanting something. An alert mind is sufficient; remaining alert for the duration of the session is quite another thing. Moreover, eyes will move. Let them so.
Q1: What are the best ways, or the best way, to determine proper meditation objects?
There is your attention... and there is your meditation object. If anything comes between the two then attention has been removed from the object. This might include thoughts about bodily phenomena. In fact, thoughts are the number one culprit for derailing ones attention from the object. You can observe bodily phenomena without thought.
Q2 : Is there any real drawbacks of meditation with a physical object or mental image as compared to the breath?
This is personal preference which can be understood further by experimentation.
As further reading I’d strongly suggest that you familiarise yourself with the five hindrances.
The Five Hindrances