[RPG] Do weapons’ damage types (P/S/B) have any effect

No.

RAW, no, weapons do not break themselves. Only damage explicitly done to objects via sundering or striking an object apply. The rules in Damaging Objects are the sum of how you can do it; if it's not there it's not there for a reason. They also don't say your weapons don't disintegrate every Tuesday, "it doesn't say it doesn't happen" is not a logical argument about a ruleset.

Furthermore, lances aren't more fragile than any other two handed wooden hafted weapon in Pathfinder. (Nor were they fragile in real life - jousting lances were specially hollowed but ones for real fighting were as sturdy as anything else.)

You can house rule that damage applies to weapons - but given the inflated damage numbers of Pathfinder, weapons will be breaking on every hit. Over level 3 hits routinely do 15+ points of damage per strike and that would explode a lance or spear or halberd every time. So then you'd need to do "half damage" or something and it becomes a logistical challenge. Note that weapons become "broken" at half hit points and provide a penalty and are completely destroyed at 0 hp. If implementing a weapon damage rule, I'd take it to a more realistic level; maybe do 1d6 damage to it on a natural 1 (or maybe a number equivalent to the hardness of the thing being struck).

While it doesn't seem to be explicitly spelled out anywhere, the Sneak Attack entry says:

Beginning at 1st level, you know how to strike subtly and exploit a foe’s distraction. Once per turn, you can deal an extra 1d6 damage to one creature you hit with an attack if you have advantage on the attack roll. The attack must use a finesse or a ranged weapon.

This is extra damage dealt by the weapon attack; therefore, it should deal damage of the same type as the weapon. As backup for this, the section on critical hits says that:

For example, if you score a critical hit with a dagger, roll 2d4 for the damage, rather than 1d4, and then add your relevant ability modifier. If the attack involves other damage dice, such as from the rogue’s Sneak Attack feature, you roll those dice twice as well.

It seems clear that Sneak Attack is extra damage added to the weapon attack, and therefore has the same damage type as the weapon.

Specifically, the Spy uses a shortsword, which deals piercing damage, so his Sneak Attack damage will be piercing also.