under Stealth, "You can move up to half your normal speed and use Stealth at no penalty. When moving at a speed greater than half but less than your normal speed, you take a –5 penalty. It’s impossible to use Stealth while attacking, running, or charging.
Breaking Stealth When you start your turn using Stealth, you can leave cover or concealment and remain unobserved as long as you succeed at a Stealth check and end your turn in cover or concealment. Your Stealth immediately ends after you make an attack roll, whether or not the attack is successful (except when sniping as noted below)."
Dimensional Slide is not stealthy in the sense that it has no circumstance modifier nor does it call for a stealth check to use, so it's not "stealthy" in that way. Like Teleport there's no mention of sound or a sound effect.
Going from cover to cover it's fine (may be a -5, see below).
The verbage about succeed means others fail at their Perception Checks vs your Stealth Check as it's an opposed check.
Dimensional slide does use 5ft of movement and that clearly does count against your total move(towards half your speed) for the round. The only hiccup is if a GM adds the teleport distance to your speed as then you are going to exceed half pretty quick. The teleport distance shouldn't be added.
Lastly you need Dimensional Agility/Dimensional Assault feat(s) to Dimension Door and attack, but that's Dim Door. While the feat specifically mentions Dim Door and Abundant Step, most home game GMs should allow other teleportation effects to work with the feats.
If you're escaping don't bother with Vanish, just slide away then use the rest of your move, then use your standard to move some more or cast a spell out of sight.
