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What you are looking for is the Degree of Success rules.
There are four degrees of success. Critical Failure, (regular) Failure, (regular) Success, and Critical Success.
There are two rules in play for determining the degree of success. One is to compare the final roll result to the listed DC. The other one is the Nat-1/Nat-20 rule.
Once the roll result gives you an initial degree of success, the Nat-1/Nat-20 rule can change that result down or up by one step.
And that applies to just about every d20 roll in the game. Even if a check doesn't list out outcomes for each degree of success, the four degrees do still exist.
There are also feats, features, and effects that can change particular type of check for specified degrees of success, such as the Rogue Evasion class feature that causes a Success on a Reflex save to be upgraded to a Critical Success.
So with that in mind:
How do Nat 1s and Nat 20s affect the following: Attack Rolls, Skill Check Rolls, Saving Throws from spells and such, and any other roll via a d20?
Yes. Pretty much every d20 roll.
I believe Nat 1s and 20s cause auto-hit (5%/95%) on Attack Rolls.
Not quite. If you flail away with a 3rd attack at -10 M.A.P against a CR +3 boss enemy, and roll a 20 - it is very possible that your final total will fall into the Failure outcome and the Nat-20 rule will upgrade that to a regular Success.
Similarly, it is technically possible that Nat-20, plus bonuses, minus penalties ends up in the critical failure result and the Nat-20 rule will only upgrade that to a regular failure.
That doesn't happen in anything other than extreme and somewhat contrived situations though. Normally if you roll a Nat-20 you are pretty certain to get a critical, and if you roll a Nat-1 it will equally certainly be a critical failure.
1 Degree success/worse on Skill Checks. (No auto-fail)
Correct.
1 Degree success/worse on Saving Throws. (No auto-fail)
Correct.
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The other one that gets hidden and forgotten about is this one:
If a feat, magic item, spell, or other effect does not list a critical success or critical failure, treat is as an ordinary success or failure instead.
So if something doesn't list a critical effect (such as Strike not listing a Critical Failure effect), it uses the same effect as the normal effect. So a Critical Failure on a Strike would do the same as a normal Failure on a Strike.
What isn't explicitly stated anywhere that I know of is that if some check result doesn't list a regular effect (like Strike not listing a regular Fail effect), then the result is that nothing happens.
There are a few things that don't list a regular effect, but do list a corresponding critical effect. So getting a regular Failure for example, would do nothing, but getting a Critical Failure would do its listed effect. I can't remember one to pull up immediately though.
If you prefer graphical representations then you might like my universal check tables.
They show you the probabilities for attack rolls, skill checks, and saves and also show you how the results of the die roll affects the outcome of each.
You can see that the '20' row jumps up one degree of success from the pattern established by rows '2' through '19,' and the '1' row drops down one degree of success from that pattern.