But what about Mario's other obstacles and enemies? Which things can be controlled, and which cannot?
There is no known method for influencing the springs (nor is there a pattern for their appearance). You get what you get.
That leaves Mario's most difficult and active enemies, the fireballs and firefoxes.
The bottom line is this: as with barrels, when fireballs or firefoxes are on the same horizontal level as Mario, they are not influenced in any way by player input or relative position (at least not in a meaningful, controllable way).
It doesn't matter if you run away from them, toward them, jump over them, or whether you're near, far, or inbetween—nothing you do affects where they move or when.
Fire-beings are only affected by player input or Mario's position under two (possibly three) conditions:
Mario's relative vertical position: Firefoxes can be at a level below Mario at any time, but they cannot use a ladder to descend to a level below Mario. If Mario is above them, and they encounter a ladder, and they decide to use it, they can only climb. This includes fireballs on the barrels screen—they can duck down the ladder when you're on the same girder that they are (frustratingly so when you have the hammer), but this is because the girders have an incline where Mario can technically be "below" the fireball while still on the same girder.
Mario's horizontal position at the time they spawn (initially when entering the screen, and upon reincarnation after being smashed by the hammer): Firefoxes will always spawn on the side of the screen opposite Mario's position, whereas fireballs on the Pans screen will leap from the oil can onto the side of the screen that Mario occupies.
It is believed by some that Mario's position relative to a ladder influences whether firefoxes will use the ladder. I haven't observed this conclusively and am currently investigating.
All other fire movement is random. In fact, depending on how you look at it, it will either be humbling or empowering, when dealing with the fireballs and firefoxes, to remind yourself of the fact that they don't even know you're there.
The truth is that the fire-beings are wandering without aim. They are confined by the standard rules governing their behavior, but are moving from here to there with no rhyme or reason. They are blind, deaf, and dumb.
Any perceived player input or positioning influence, aside from the aforementioned, is a figment of the player's imagination. It's definitely not going to seem that way, and you're going to trick yourself into thinking that you've got their number, but you don't.
The (horizontal) movement of the fire is only influenced by player input inasmuch as player input acts as a random number seed for what the game code outputs.
In other words, your actions (or lack thereof) DO make a difference in what the fire-beings do, but not in a patternistic manner, or in any other way that can be favorably exploited or predicted.
Some players do not believe this, and there are moments when I myself will swear on my own eternal soul that player input DOES have an effect, but I can't explain or support anything specifically, and must stand guard against my own cognitive biases that are leading me to this conclusion.
For now I surrender to the current state of thought on the matter: analysis of the original Donkey Kong source code by minds and players more advanced than I supports the conclusion that "no player influence" is indeed the physical law of fire-beings. It's possible that this could be overturned at any time by a new discovery—things are still being found in the source code, and in practice, to this day—but as of now, this is the prevailing and accepted theory.
The following will be largely repetition of the above, but if the reader would like this information corroborated by a respected master player, I will quote Dean Saglio on the issue (who, by the way, manages to offer an extremely useful playing tip at the same time):
"Strictly speaking, the game code suggests that the fireballs are not controllable by the player with one minor exception. Any appearance that user input has any influence over the fireballs is just coincidence. The one exception is that fireballs cannot go down ladders unless Mario's vertical position is below the fireball -- so there are many places where taking one step down onto the top rung of a ladder will enable a fireball to climb down and out of the way.
However, you will find that all top players do a TON of things in an attempt to constantly control or influence fireball movement at all times. We are all convinced that such techniques have a helpful effect or else we wouldn't be doing it. In addition, the specific techniques used seem to vary from player to player even among the best players, so the best bet is to just keep experimenting until you find some things that seem to work for you."
(Remember that "top rung" tip. Trust me, it will come in handy.)