Hide in Plain sight (HiPs) – does it work if I cast light on theself

1. How does this work against creatures with darkvision considering they can see perfectly in dim light?

It works exactly the same for several reasons. Firstly the Shadowdancer isn't even ~in~ the area of dim light, they're standing in plain sight so normally anyone could see them. Secondly the shadows aren't what are hiding the shadowdancer, the shadows are the required medium for them to use their (Su) - Supernatural ability; note the text:

As long as she is within 10 feet of an area of dim light, a shadowdancer can hide herself from view in the open without anything to actually hide behind.

The shadows aren't hiding the shadowdancer, they're just allowing the shadowdancer the chance to use their supernatural ability, the shadows are like a focus for a spell.

2. How should I handle the being 10 ft near dim light?

For any general ability and/or targeting of supernatural things the character needs to be able to see (ie have line of sight) to the target/object/etc in question. So that is how I would suggest running it. If the character is standing by a 20' wall and the shadow is on the other side of the wall, it wouldn't work. If however there is a window in the wall then it would.

3. Since it specifically mentions that a shadowdancer cannot use her own shadow, does this mean she can use someone else's shadow even though there is no dim light nearby?

Yes. A shadow is a shadow is a shadow; unless it's the shadowdancers. Since the ability states they can't hide in their own shadow it's a pretty safe conclusion then that every other shadow is fair game. Otherwise why on earth are they called shadowdancers?

The actual size of the shadow becomes important then (blades of grass was mentioned in another question for 3.5) and I'd require that the shadow be of at least one size category smaller than the character.

No, you don't need to spend a move action to hide, and you can hide as part of a full attack.

First. The rules say movement is the usual method of activation:

Action

Usually none. Normally, you make a Hide check as part of movement

But that's not a necessity. That's because we also have the following rule:

Check

Your Hide check is opposed by the Spot check of anyone who might see you. You can move up to one-half your normal speed and hide at no penalty. When moving at a speed greater than one-half but less than your normal speed, you take a -5 penalty. It’s practically impossible (-20 penalty) to hide while attacking, running or charging.

So, hide while attacking all you want, you just take a teensy tiny -20 penalty on your check.

As for Hide in Plain Sight and actions, all HiPS does is change the conditions under which you can hide. While Supernatural abilities are activated as a standard action unless otherwise noted (and HiPS is a Supernatural ability), HiPS is generally understood to be a "passive" modification to the character's abilities, not requiring specific activation. I'd compare it to the Paladin's Divine Grace when it comes to this.

TL;DR version: HiPS inherits the actions from normal hiding, and hiding can be done as part of the action of attacking.