Yes to all three, though it took me a while to figure out and a slight interface quirk almost made me give up before I figured it out.
It will increment dates just fine.
To change the increment, enter the first two numbers of the desired increment in adjacent fields. Then with both fields selected do a fill.
To get the pop-up menu that lets you fill, you first select the field or fields you want. Then you tap and hold until the menu shows up. The quirk, if you don't hold for at least a second after the menu shows it goes to entry mode. Or sliding your finger off the field before releasing seems to work too.
Also, when you select a date field and accidentally go into entry mode, it resets the field to blank and you have to reinter the date. Very annoying, until you learn to just hold the tap a little longer.