There's one issue I particularly find strange, namely that Dampen Presence allows to make stealth checks against blindsight, but blindsight including
So since these specific forms of "blindsight" are out (which is what I understand them to be given the text) which other form of blindsight are trying to evade? Magical aura?
Also, how would such an opposed stealth check look like? Unmodified, straight up stealth check against perception? Would the carefully hidden trait help?
1. Normally, blindsight or blindsense can prevent you from using Stealth in situations where you otherwise could. Dampen Presence prevents blindsight and blindsense from being an issue. If the creature in question can sense you via other means that would prevent the use of Stealth, Dampen Presence won't be enough to let you Stealth by itself. It also doesn't negate any other benefits of blindsight or blindsense--just their use to prevent you from using Stealth.
2. The check would look like any other Stealth check to hide from someone.
Scent and Tremorsense are specific abilities as is Blindsight (and in a way vision, although it is assumed that 'everything' has it and only exceptions are noted.)
Dampen Presence will work again ALL forms of Blindsight or Blindsense regardless of the particular flavor they may have. A creature with Blindsight from extraordinary sense of smell or from echolocation is effected by this feat. Now, if the creature also have a special ability Scent or a Tremorsense they may still not be able to use stealth against them, but if you have 'scent based blindsense' but not the actual 'scent' special ability, Dampen Presence works just fine.
So it's in reference to a creature having a specific ability, such as scent, echolocation or something like that rather than a more general blindsight.
And the purpose of stealth being making him unable to know your location?
I understand that blindsense still permits you to have concealment (50% miss chance) if you are not otherwise detected. He just knows your (exact) square?
Let's take the example of a dragon, which you invisibly and stealthily approach. Once you get within 60 feet from it you stealth with dampen presence.
Do you have to hide behind something to stealth against blindsense? Do you still get +20 due to being invisible?
Mirror image still works though because as a figment it messes with his perception, not just vision, correct?
Normally blindsense and blindsight absolutely negate stealth: no perception roll is required to notice and locate creatures using those senses.
The feat takes that away, and makes 'stealth' against those creatures work just like they didn't have it, but it doesn't do anything else. It doesn't remove their blindsense or make you invisible to it, it pretty much just removes the auto notice.
Your specific questions a 'deep enough in the weeds' that there is going to be considerable GM interpretation. I don't believe that there are detailed enough rules to give a definitive answer, and I could see logical interpretations on both sides.
Mirror image for example mentions that if you can't see it has no effect. It doesn't address a situation where you CAN see, but also have extraordinary senses that make you not require being able to see to function. Personally I would assume that blindsight would overcome mirror image whether or not the creature using the blindsight power was also blind, but strictly by the text of the spell unless the attacker is blind mirror image functions fine against something with blindsight.