I know that in the classic example of Rod of Ruin, I can activate it immediately after casting it and even if my opponent destroys it, the 1 point of damage stays on the stack. What about something like Sneak Attack? Can I cast it, retain priority, and bring in one or more creatures via the enchantment BEFORE my opponent has an opportunity to destroy it?
You always retain priority after something you control resolves on your turn, so absolutely, but you don't even (okay generally) have to do this. In point of fact, you can activate Sneak Attack's ability while something that would destroy it is on the stack - you can easily respond to a Disenchant aimed at it by activating the ability to put a Mystic Snake onto the battlefield and counter that spell.
Yes. Outside of Stifle effects and the likes of Vendilion Clique, there isn't anything your opponent can do to stop a resolved Sneak Attack as long as you have red mana available.
Rod of Ruin's and Sneak Attack's abilities differ notably in that—
Rod of Ruin's ability requires a target to be chosen as the ability is activated (C.R. 114.1c, 601.2c, 602.2b), so that that target will be known to all players after that ability is activated and before it resolves (see also this thread), whereas
for Sneak Attack's ability, the choice of whether its controller will put a creature card from hand onto the battlefield, and if so, which, is not made until that ability resolves (C.R. 608.2d, 109.5), so without any other player knowing what that choice will be before it resolves.
In both cases, players can cast instant spells (including instant spells that would remove Sneak Attack or Rod of Ruin from the battlefield) in response to Sneak Attack's or Rod of Ruin's ability (that is, while it's on the stack) as they can to any other activated ability that goes on the stack (C.R. 116.7, 116.1a). And in both cases, the ability will get to resolve even if Sneak Attack or Rod of Ruin, respectively, is no longer on the battlefield (C.R. 112.7a). But no player has priority to cast such spells while that ability resolves (e.g., for Sneak Attack's ability, between the time "you choose" whether to put a creature card from hand onto the battlefield, and if so, which, and the time it enters the battlefield, as the case may be) (C.R. 116.2e, 116.1a) (see also this thread and this thread).
You always retain priority after something you control resolves [...]
That's incorrect. You get priority back after you have cast a spell or activated an ability if you had priority before doing so. But after any object on the stack has resolved, the active player gets priority, no matter who controlled the object. In this specific case, the caster and the active player are the same, so the answers given are still correct.
Edit: Correction.
I̟̥͍̠ͅn̩͉̣͍̬͚ͅ ̬̬͖t̯̹̞̺͖͓̯̤h̘͍̬e͙̯͈̖̼̮ ̭̬f̺̲̲̪i͙͉̟̩̰r̪̝͚͈̝̥͍̝̲s̼̻͇̘̳͔ͅt̲̺̳̗̜̪̙ ̳̺̥̻͚̗ͅm̜̜̟̰͈͓͎͇o̝̖̮̝͇m̯̻̞̼̫̗͓̤e̩̯̬̮̩n͎̱̪̲̹͖t͇̖s̰̮ͅ,̤̲͙̻̭̻̯̹̰ ̖t̫̙̺̯͖͚̯ͅh͙̯̦̳̗̰̟e͖̪͉̼̯ ̪͕g̞̣͔a̗̦t̬̬͓͙̫̖̭̻e̩̻̯ ̜̖̦̖̤̭͙̬t̞̹̥̪͎͉ͅo͕͚͍͇̲͇͓̺ ̭̬͙͈̣̻t͈͍͙͓̫̖͙̩h̪̬̖̙e̗͈ ̗̬̟̞̺̤͉̯ͅa̦̯͚̙̜̮f͉͙̲̣̞̼t̪̤̞̣͚e̲͉̳̥r͇̪̙͚͓l̥̞̞͎̹̯̹ͅi͓̬f̮̥̬̞͈ͅe͎ ̟̩̤̳̠̯̩̯o̮̘̲p̟͚̣̞͉͓e͍̩̣n͔̼͕͚̜e̬̱d̼̘͎̖̹͍̮̠,͖̺̭̱̮ ̣̲͖̬̪̭̥a̪͚n̟̲̝̤̤̞̗d̘̱̗͇̮͕̳͕͔ ͖̞͉͎t̹̙͎h̰̱͉̗e̪̞̱̝̹̩ͅ ̠̱̩̭̦p̯̙e͓o̳͚̰̯̺̱̰͔̘p̬͎̱̣̼̩͇l̗̟̖͚̠e̱͉͔̱̦̬̟̙ ̖͚̪͔̼̦w̺̖̤̱e͖̗̻̦͓̖̘̜r̭̥e͔̹̫̱͕̦̰͕ ̗͔̠p̠̗͍͍̱̳̠r̰͔͎̰o͉̥͓̰͚̥s̟͚̹̱͔̣t͉̙̳̖͖̪̮r̥̘̥͙̹a͉̟̫̟̳̠̟̭t͈̜̰͈͎e̞̣̭̲̬ ͚̗̯̟͙i͍͖̰̘̦͖͉ṇ̮̻̯̦̲̩͍ ̦̮͚̫̤t͉͖̫͕ͅͅh͙̮̻̘̣̮̼e͕̺ ͙l͕̠͎̰̥i̲͓͉̲g̫̳̟͈͇̖h̠̦̖t͓̯͎̗ ̳̪̘̟̙̩̦o̫̲f̙͔̰̙̠ ̹̪̗͇̯t͖̼̼͉͖̬h̹͇̩e͚̖̺̤͉̹͕̪ ͚͓̭̝̺G͎̗̯̩o̫̯̮̟̮̳̘d̜̲͙̠-̩̳̯̲̗̜P̹̘̥͉̝h͍͈̗̖̝ͅa͍̗̮̼̗r̜̖͇̙̺a̭̺͔̞̳͈o̪̣͓̯̬͙̯̰̗h̖̦͈̥̯͔.͇̣̙̝
In both cases, players can cast instant spells (including instant spells that would remove Sneak Attack or Rod of Ruin from the battlefield) in response to Sneak Attack's or Rod of Ruin's ability (that is, while it's on the stack) as they can to any other activated ability that goes on the stack (C.R. 116.7, 116.1a). And in both cases, the ability will get to resolve even if Sneak Attack or Rod of Ruin, respectively, is no longer on the battlefield (C.R. 112.7a). But no player has priority to cast such spells while that ability resolves (e.g., for Sneak Attack's ability, between the time "you choose" whether to put a creature card from hand onto the battlefield, and if so, which, and the time it enters the battlefield, as the case may be) (C.R. 116.2e, 116.1a) (see also this thread and this thread).
EDIT (Mar. 27, 2019): Correctness edit.
That's incorrect. You get priority back after you have cast a spell or activated an ability if you had priority before doing so. But after any object on the stack has resolved, the active player gets priority, no matter who controlled the object. In this specific case, the caster and the active player are the same, so the answers given are still correct.
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