I have found that trying to maintain an exact balance was just impossible. As you said, it ends up being ridiculous: if I help an old person to cross the road, do I have to steal her bag right after that?
So after a long time trying to play this kind of character, I realized that the important part was not to maintain an hypothetical balance (which might not even exist in the first place), but to avoid extremes.
For example:
You don't want the evil demons to eradicate the Good Guys. But you don't want either the Good Guys to eradicate the evil demons.
Total war is bad. World peace too.
Civilization leading to Nature's extermination is bad. But Nature overwhelmingly destroying civilization is bad too.
It ends up being more fun to play, as you don't have to worry about any single action, only the grand scheme of things.
And I believe it is closer to the intended concept, as by making sure that both sides of the coin always exist, you are really working towards Balance.
(it will also makes you develop a "Neutral Mind", which is always a good thing)
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?
Best Answer
You ask, can specific references be found as to what constitutes a good or evil act in 5e.
Ideals
Ideals are one area to look. In the PHB (p124) under ideals, it says:
That section goes on to list ideals for the various backgrounds. Many of the ideals are tagged with good, evil, lawful, or chaotic. I've listed the good and evil ones below.
Based on the descriptions, good is associated with beauty, charity, generosity, the greater good, and respect. Evil is associated with greed, mastery, might, power, and retribution. Although the items are generally more attitudes than specific acts, you can make some extrapolations based on the ideals.
Good ideals:
Evil ideals:
Dragons
The descriptions in the Monster Manual describe the chromatic dragons as evil and the metallic dragons as good.
The chromatic dragons are described as aggressive, gluttonous, vain, powerful tyrants feared by all creatures, driven by greed, lust after treasure, greed colors their every scheme and plot, believe that the world's wealth belongs to them by right.
The metallic dragons are described as pretty much the opposite, as inclined to actively oppose tyranny, lend a helping hand in unseen ways, befriend a lost child, a wandering minstrel, or an innkeeper.
In particular silver dragons are described as:
Other Monster Manual Examples
There many more Monster Manual examples of good and evil creatures, particularly evil ones. Of interest might be the demon and devil entries, the hags, the incubi/succubi, and others.