[RPG] How to an evil outsider be permanently destroyed

This seems to be a case of the creature description being based on a development version of a spell that was changed before publication, but the bloody skeleton's ability wasn't updated to match. Or the design notes for the bloody skeleton read something like "or 0 hp on blessed ground" and the developer that actually wrote the text didn't do due diligence in translating that into the correct spells. However it happened, it's an error rather than a mysterious rules interaction.

Go with your gut, and change it to "the area of a consecrate or hallow spell". Pathfinder and d20 system games in general expect GMs to make their worlds believable rather than to slavishly follow rules into weird contradictions and twisted corner cases that break suspension of disbelief, so you're entirely in the right to change this in your own games. If you're a player in another GM's game, suggest the fix as a sensible thing to do, but leave it up to your GM.

but how would performing these almost surely evil acts affect how my character is expected to play the game from then on?

To me this is the most important thing you asked. The answer is: Not At All.

Your character appears to have a well-defined backstory and personalty. Those things define how your character should act. Alignment should reflect those, and that's all. Your alignment changing because of your actions should NOT affect how your character acts, as the alignment just shifted to reflect what you were already doing.

Don't fall into the trap of going "oh I got revenge and the DM made me evil, so now I'm throwing out my character's persona and going supervillan."

Alignment is there to help people who don't have a clear personality for characters, provide game mechanics for things like Smite Evil, and provide some restrictions that we could probably do without. It's not there to act as a straightjacket on what is a well-developed character.

Now, the actual act of getting revenge and fallout from that might cause your character to behave differently. That's RP and entirely in-bounds. If the party treats you differently afterwards that is going to have an effect, but the alignment system doesn't reflect that very well. (Parties can shun good aligned people too.)

How much should other players be aware of my plans?

Depends on the players, really. Are they players who will be okay with some manipulation and secrecy? Or are they players who really value team cohesion and won't appreciate being led around for a secret potentially evil plan?

There's not a good way for me to answer that, as you know them better than I do. I would say that your character probably won't tell them more than necessary unless he's sure that telling them would make them more willing to help.

How careful should I be about players' reactions to having my PC manipulate other PCs?

Same thing as above, really. Some players would have no problem with it at all, others would be rather unhappy.

If my character does manage to enact his revenge (and he may never finish doing so), assuming the actions are metaphysically evil (e.g., knowingly killing a good-aligned entity, etc.), does this make my character evil? What about the other PCs - are they evil? If not, do they turn on me?

While killing a Good creature like an Angel could be called an outright evil act, one evil act doesn't necessarily make you evil. Mortals almost never follow alignments perfectly, they're a mix of different acts and it's more the trend that determines it.

The other PCs may or may not have their own alignment trouble, depending on the situation. How much do they know about why they're helping you? How good are they usually?

And no, if they're helping you do this knowingly and your alignment shifts because of it, they shouldn't just automatically turn on you. That wouldn't make sense, would it?