Here's what you're trading for the alertness feat if you don't take the abil score upgrade and instead take the feat:
- +1 to damage
- +1 to hit
- +1 to AC
- +1 to Dex saves
- +1 to Dex checks
- +1 to init
The question then, ultimately, becomes, is +5 init (net +4), no surprise and no advantage on attacks against from hidden opponents worth it.
The latter two features are very dependent on your campaign and DM. Surprise should be a fairly frequent occurrence on both sides of combat, but that still depends on how willing your DM is to allow it or stage his monsters to have them obtain it (and kind of how cautious your group tends to be). Monsters attacking from hidden is another one that really depends on your DM, you group and how well you guys are at nosing out monsters from the shadows.
So ignoring those two items since they are DM and campaign dependent (and thus can't be weighted objectively), the question then becomes, is +1 to Dex more or less beneficial over 4 levels than +5 to init.
For pretty much any character other than the Assassin Rogue, I'd probably argue that they should take the stat upgrade early, the feat at L8 and then the last stat upgrade. However +5 init has amazing synergy with the Assassin Rogue and you're going to get a ton of mileage out of it.
If you're already at 20 Dex (if you managed to start at 20, or 18 and are at L8), then there really isn't any stat upgrade that is worth taking over this feat (though there may be better feats). Rogues are skill monkeys, yes, but +5% in a single ability set of skills is not worth the +25% to initiative and other features o this feat. The only one that would be worth considering at all is Con for the extra HP, but it's really not that many and you're better of boosting your damage significantly.
Take the feat at L4 (or even better, play a human and take it at L1).
Option 1 is the correct answer here
Ability checks always add up d20 + ability modifier + situational modifiers, as long as this is greater or equal to the DC then you succeed at the task. This applies to all ability checks for all ability scores.
In terms of a natural 20, this is entirely up to your DM from what I can find. For the group that I DM I allow success on the task if it seems fairly reasonable, but for instance a player trying to punch through a dungeon wall would never succeed with fists alone (without special fists too).
Generally if the task is impossible, the DM should directly say that you make the attempt and fail, no roll necessary.
Best Answer
The modifier is directly linked to the stat, so a STR of 19 has a modifier of +4. That just happens automatically. STR based attacks will therefore benefit from the higher modifier. Skill checks will improve if they use the STR modifier as it has improved.
Basically, the modifier instantly increases to match the new total of the stat then that flows over into attack rolls, damage rolls and ability checks/saves that use that modifier.
From the SRD: