In R/S/E, only the timing of actions can influence random factors. Holding down a button, when there is no visible effect, does nothing. Because many dialogs run at a fixed rate on fast speed, rather than prompting for button presses, it is vital to set the text to mid speed in order to control the rate of text in order to make luck-manipulation better.
An attack miss is determined before the attack message "(Pokémon) used (attack)".
When both Pokémon have the same speed, all things being equal, the one going first is decided when the attack is selected in the menu.
When using Quick Claw (1/4 chance to go first if slower, all other things being equal), the chance is determined before the menu appears.
Paralyze effect (unable to move) is determined just before the attack is attempted, and confusion effect (self-damage) is determined after "(Pokémon) is confused!"
Added effects of attacks (flinch, stat boost, confusion, paralyze, etc.) are determined after the hit, after critical hit and super effective messages if any.
In R/S/E, trainers that change the direction in which they look are influenced by running near them. It is required to walk or bike by them to avoid triggering a battle.
In R/S/E, wild Pokémon encounters are determined when you step into a patch of grass.
Enemy Pokémon moves are determined when the attack is selected.
Trainer moves are determined before the menu appears.
Smart trainers will always use stronger attacking moves over weaker ones.
Smart trainers will always try to use moves that knock out your Pokémon if it is possible.
Smart trainers may also do things like use potions or switch to a Pokémon that resists your last attack.
The first Pokémon is always fixed.
Smart trainers will always bring in a Pokémon that can knock out your Pokémon if it is possible, and will always avoid bringing in a Pokémon that is weak to yours.
Pokémon order can only be manipulated from what your current Pokémon is, not from luck-manipulation.
Pokémon personal stats
Personal stats (IVs, nature, ability, shiny, Pokérus, etc) for the starter Pokémon are determined when you obtain the Pokémon.
Personal stats for wild Pokémon are determined on encounter.
Zigzagoon's Pickup ability is determined on the fadeout after each battle.