Your second question is separate from the first. May I suggest you create a separate thread for that question, so that others can easily find it when searching for this answer.
I do have a solution for you, which I use in most of my libraries, and which I will share in the new thread.
For the first question, there a many ways to accomplish this. If you want to do this in simple Javascript, the following works:
var e = entry();
var dt = new Date(e.field("DateField"));
dt.setDate(dt.getDate() + 1);
e.set("DateField", dt.getTime());
The above still works when at the end of the month or year.
One advantage with using Javascript is that it adds flexibility when/if your requirements change.