How many days a month can you see a moonrise during daylight? For about 15 days before a Full Moon the moon rises during the day, and for about 15 days after a Full Moon the moon sets during the day. I think it might be possible at full moon for the moon to rise just before daylight ends, and to set just after the next daylight starts, on some occasions.
However not all of these rises/sets will be visible. Around dark moon, moonrise is lost in the rising sun, and moonset is lost in the setting sun.
Hence my question, how many can you actually see?
I know this depends on a lot of factors, so a range would be sufficient.
I think this is equivalent to asking, what elongation of the moon is needed for the moon to be visible? When both are above the horizon?
It's too hard for me to come up with reliable figures for the amount of visible moonrises/moonsets during daylight, but I can answer

what elongation of the moon is needed for the moon to be visible?

It will depend on the conditions, but according to the Wikipedia article about the Islamic calendar, the Moon may already be visible 17 hours after the conjunction with the Sun:

But, the lunar crescent becomes visible only some 17 hours after the conjunction, and only subject to the existence of a number of favourable conditions relative to weather, time, geographic location, as well as various astronomical parameters.

The synodic month is about 708.734 hours, so a period of 17 hours corresponds to an elongation of $360^{\circ} \cdot \frac{17}{708.734} = 8.6^{\circ}$.