[RPG] Can a Half-Elf choose the same Ability Score for the +1’s of Ability Score Increase

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).

Yes, the Star card can permanently increase your ability scores beyond the normal limit of 20.

The Player's Handbook, on page 15 in the section about applying ability score improvements, states the following rule:

Some of these features allow you to increase your ability scores, either increasing two scores by 1 each or increasing one score by 2. You can't increase an ability score above 20.

Note that, specifically, this does not say "you cannot have an ability score above 20"; it states that you cannot use ability score improvements to increase an ability score above 20 - the rule only has importance at the moment you are altering your stats.

Following standard "specific beats general" interpretation, the Star card offers you an ability score improvement which specifically clarifies that it can, in fact, raise your ability score above 20, up to 24. Having applied the benefit of the card, if your ability score is now 20 or higher, you can't improve it further with a normal ability score increase, as that's subject to the general rule that you can't increase the score above 20. But there's no rule that states that, if you have legally increased your score past 20 by some means, it has to go back down again. It's not really so much a "cap" as a threshold - it's difficult to cross but you're not forced back if you do it.