[RPG] Does sleep immunity mean you don’t need sleep

The d20PFSRD collates the relevant bits conveniently for you here.

Optional but RAW:

You might also consider using a variant rule where characters who do not get a full night’s sleep may suffer the effects of fatigue. If a PC does not get at least 6 hours of sleep, she must make a DC 15 Fortitude save or be fatigued and take a –1 penalty on all other checks and saving throws against sleep effects. A second night without sleep requires another DC 15 Fortitude save. A failed save results in the character becoming exhausted and the penalties increasing to –2. A third failed save on the next night increases the penalties to –3.

and from the Armor Check Penalty section:

A creature that rests or sleeps in medium or heavier armor gains the fatigued condition if they do not possess either the Endurance feat or some other class or race ability which allows them to sleep in such uncomfortable gear.

As for inclement weather and the like, the weather section doesn't have instructions on how much it disrupts sleep but I'd think that anything that overcomes your +10 Perception DC when asleep would wake you up.

Continued forced march or hustling deals non-lethal damage if you fail the required constitution check. Normally, receiving non-lethal damage this way would cause you to become fatigued, which would be prevented by extreme endurance. This would not, however, prevent the non-lethal damage from piling up and eventually putting you out of commission.

As for rules about sleep deprivation, this answer contains this link to the SRD that points to a rule introduced in an adventure path that gives cumulative penalties due lack of sleep. This is the closest thing to RAW that I can find about it, as the source books are pretty ambiguous about the issue and the only (official) consequences of skipping rest is that you do not receive the several benefits of resting.

As requested in comments, I have compiled a list of things that would be prevented by extreme endurance in the level 5 and in level 10:

After level 5, extreme endurance prevents you to become fatigued. This covers exhausting physical activity, as noted before, the aftereffects of barbarian rage, and certain spells and special abilities (a example here, the complete list is way longer). Note that extreme endurance only protects you from fatigue, and damage and all other negative effects that come alongside the fatigued status apply as normal. Also, this incidentally protects you from becoming exhausted due fatigue stacking (doing something that gives you the fatigue status while already fatigued). Also note that even if you manage to get yourself exhausted without being fatigued before, resting for 1 hour removes the exhausted condition and gives you the fatigued condition instead, which is rendered null and void by extreme endurance.

After level 10, extreme endurance prevents you to become exhausted. As the more usual way of acquiring this state is already covered by the level 5 version, this really limits the usefulness of the level 10 version, as its only real use is to prevent the effects of the few spells and special abilities that bestow that state directly (like this and this). As before, you are only protected from exhausted and fatigued status caused by those abilities, other effects will still affect you.