I need some clarity about the difference in the roles played by one's distance modifier and weapon skill level.

The game clearly states that Perception affects your character's ranged combat distance modifier. So my question is this:

Does Perception / distance modifier only affect the distance you're able to attain with a ranged weapon, and the skill level (small guns, energy weapons etc.) only affect the accuracy?

Example: If you have a sniper rifle with 1 Perception but 150% small guns, does this mean you won't get the full range benefit out of the weapon, but when the target does come into range, then you'll be extremely accurate?

And the inverse: If you have 10 Perception but only 50% in small guns, do you get the full range of the sniper rifle, but your accuracy will be problematic?

Or, do the two factors overlap with each another: Meaning, anybody can get the full range benefit out of the sniper rifle, but both Perception and weapon skill level will affect accuracy? Can you for example get increased range by increasing your weapon skill without increasing your Perception?

I don't know if this is making sense. Could somebody please help explain?

Are you sure you didn't make a typo somewhere in your post? The calculations don't all appear to be consistent. Or maybe I'm just not understanding properly.

I don't see a typo, which part do you think is inconsistent?

OK, let me plug in some examples into those formulas to see if I understand correctly. Character as follows:

Perception = 7
Small Guns = 64%

Effective range: 6 [(7 * 2) - 8]

Target within effective range:
Chance to hit: 120% [64 + (7 * 8)]

Target outside effective range (12 hexes away.):
Chance to hit: 56% [64 + ((7 - 2) * 8) - (12 * 4)]

Target outside effective range when weapon long range perk = Yes. (12 hexes away.):
Chance to hit: 96% (64 + ((7 - 2) * 16) - (12 * 4))

That's almost correct.
Effective range is there only on weapons with the long range perk. On normal guns the CTH has just one formula

Chance to hit: 56% [64 + ((7 - 2) * 8) - (12 * 4)]

If the weapon HAS the long range perk the CTH is either

Target within effective range:
Chance to hit: 120% [64 + (7 * 8)]

OR

Target outside effective range
Chance to hit: 96% (64 + ((7 - 2) * 16) - (12 * 4))

Do note that in those formulas other modifiers (like ammo modifiers) are disregarded for the sake of simplicity. But you can easily test it ingame. As an example, the plasma rifle has the long range perk. Let's say that you're outside the effective range and your CTH is 50%. You pop a mentat for PE+2 while still staying outside the effective range. The CTH should now be 82%.

OK, so Perception certainly makes a huge difference to ranged accuracy. It would then be well worth it to take a Perk like Sharpshooter, right?

It's better to improve your CTH by consuming Mentats. There's no perk cost, it's basically for free and the Mentats effect lasts long. Let's say that you're trying to clean the Military Base. You pop 2 mentats for +4 PE, that's a cheap CTH gain and you can clean all of Military Base with it.

The main benefit of sharpshooter is that the +2 PE bonus is cumulative with the 10PE stat cap. On a PE10 character (or PE6 +4 from Mentats) and the sharpshooter perk, you're having accuracy bonuses as if you had PE12. Sharpshooter is therefore useful for some theoretical CTH maximization purposes, but it does cost you a perk. In a normal game, you should be able to get to 95% CTH without wasting a perk on sharpshooter.

In FO1 it's even possible to clean the Military Base with the starting big guns skill (~40%) by upping your PE to 10 with mentats and using a rocket launcher and having sneak turned on. Rocket Launcher has the long range perk AND you're just firing at the torso, so no aiming penalties. Considering that Military Base is supposed to be an endgame location in FO1, it's a good showcase of PE's power in Fallout.

I'm not a huge fan using chems, unless in a desperate situation. Like you said, the effect is temporary, but then it has side-effects that you have to deal with, as well. It's also a consumable, which means you can run out, so how can you depend on them for something as fundamental as character stats?

By the way, do you know if one's skill level also affects damage, or is it only accuracy? (In the cases of both ranged and unarmed / melee.)

Weapon skill affects only accuracy, not damage. There's a slight indirect dependence between CTH and critical chance. When you make a critical roll, there's a small random additive bonus to your crit chance. The random bonus is between 0 and CTH/10. With CTH 95% that would be between 0-9.5. With CTH 50% the bonus is slightly lower, 0-5. So with higher CTH, you'll score slightly more critical hits. But the difference in the crit chance is small, basically negligible. Fundamentally, your skill and your CTH have no effect on the damage.