How many days a month can you see a moonrise during daylight?
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 $360^{\circ} \cdot \frac{17}{708.734} = 8.6^{\circ}$.