Only effects requiring attack rolls yield critical hits

The splash damage of an alchemist's bomb isn't increased on a critical hit because the alchemist didn't make an attack roll to hit the splashed creatures with the splash damage. The rules for Critical Hits say

When you make an attack roll and get a natural 20 (the d20 shows 20), you hit regardless of your target's Armor Class, and you have scored a "threat," meaning the hit might be a critical hit (or "crit"). To find out if it's a critical hit, you immediately make an attempt to "confirm" the critical hit—another attack roll with all the same modifiers as the attack roll you just made. If the confirmation roll also results in a hit against the target's AC, your original hit is a critical hit. (The critical roll just needs to hit to give you a crit, it doesn't need to come up 20 again.) If the confirmation roll is a miss, then your hit is just a regular hit.

A critical hit means that you roll your damage more than once, with all your usual bonuses, and add the rolls together. Unless otherwise specified, the threat range for a critical hit on an attack roll is 20, and the multiplier is ×2.

Emphasis mine. The rules for when a creature takes the action Throw Splash Weapon include

A hit deals direct hit damage to the target, and splash damage to all creatures within 5 feet of the target.

The alchemist made an attack roll to hit directly the target; the alchemist didn't make an attack roll to hit the creatures within 5 ft. of the target. Thus even on a critical hit the alchemist deals his normal splash damage (although he still deals extra damage to the target hit directly).

A trample victim is not normally rendered prone

Borrowing from this question, below I've modified the steps it presents to jibe with how I think the trample process works. Be aware that the combat maneuver overrun is plagued with inconsistencies and lacks clarity, so the GM must address some parts of it because the rules don't. Nonetheless, there is some designer input on this particular issue: in a 2010 thread Paizo creative director James Jacobs says that trample victims "aren't automatically knocked prone."

Steps in the Trample Process

The creature possessing the extraordinary ability trample takes a full-round action to move—likely only up to its speed, but ask the GM—, its path occupied by at least one foe of at least one size category littler than the creature.

1. The creature enters a square the foe occupies, provoking an attack of opportunity from the foe. The GM may rule that a creature possessing the feat Improved Overrun doesn't provoke an attack of opportunity from the foe.

   • A foe that makes this attack of opportunity suffers a –4 penalty on the attack roll and is dealt the creature's slam damage + 1½× the creature's Strength modifier.
   • A foe that doesn't (or can't) make this attack of opportunity makes a Reflex saving throw (DC 10 + ½ the creature's Hit Dice + the creature's Strength modifier). Failure means that the foe is dealt the creature's slam damage + 1½× the creature's Strength modifier. Success means that the creature's dealt half damage.

2. The creature continues its movement, restarting the process upon entering a square of a foe that this round has not suffered the creature's trample damage.

A creature can enter squares occupied by the same foe multiple times during a round (likely provoking an attack of opportunity only the first time, but ask the GM). However, a creature can deal its trample damage to a foe only once per round this way.

What This Means

Because there is no combat maneuver check made, there's no chance for the creature to fail an overrun attempt (so, for example, the creature can't be returned to its starting square) nor for the creature to succeed on the overrun attempt by at least 5 (which would render the foe prone—this renders the feat Greater Overrun largely pointless for the creature, too).