I've never gone Nashor's first, neither have I gotten Lich Bane in lane. The only time I actually bought that was to replace boots... (as it gives movespeed).
I just don't feel like the attack speed from Nashors is that useful when going for kills. I mean, it's good, but not enough to replace your burst. But the Cdr is definitely needed, so I almost always take it second.
I usually go full burst + Nashors. I got Abyssal once against a Fizz, but that was pretty much it... from my perspective, flat AP is at least as good on her as other more specialized items. Her combo still uses AP rather efficiently, and Q + R is up ever 3-4 seconds.
Building flat AP makes her pretty strong early. Can I have a reason as too why I shouldn't take Zhonya's? Tank stats are never bad, as too migitate AD poke.
Another thing is... Rylai's Crystal Scepter is now a reasonably strong item on mages, due to giving them 100 AP, which is now the same as Zhonya's or Luden's Echo. And specifically for Diana, it makes her stickier due to not have to use E for the slow, which saves mana.
About mana...
Mainly, I'm running out of mana after 1 long fight (2-3 Q+R), or 2 combos. Which is pretty good when you look at it, but it means you have to recall pretty much right after. Especially because after/if I get an early lead, I'm jumping on them every time I can land my Q, and they aren't in range of their tower.
And my build + playstyle is probably why I have mana problems. My build gives me full AP, meaning multiple combo's are more effective/ quicker than a whole bunch of auto's which use less mana. Add that to using the full combo any time I can...
Also, for the tip: Doesn't work unless you can get off auto's. Also, they're usually a bit more cautious of you, which makes it harder to land a Q. Surprise is better than a charged passive, in my opinion.
And... I still have some trouble playing against, say, an Annie, due to not being able to dodge the Q, and frequent auto's to poke. By the time I get 6 I'm usually too low to all in, and it's even harder to farm cuz once they get their ult, it's easier for them to chain a stun into an ult into a kill.
Again, Nashor's is core because it perfectly synergizes with the entirety of her kit. Her passive's empowered AA is her single highest damage ability. The only way to get away with flat AP is if you can reliably 100-0 anyone with Q+R+W+R (hint: you can't do it early, which means you can't get fed enough to do it late). Nashor's gives you a bit of sustained damage to weave between combos as well as enough extra AS that you can get your passive off quickly even if you didn't prep it. The AP boost it got recently only makes it more attractive.
As an aside, Nashor's is highly gold-efficient even if you have no AP from any other sources, and only grows better as you gain more. Zhonya's is barely gold-efficient; Luden's flat out isn't.
On another note, Nashor's also helps with mana problems. Those AAs hurt even early, and you can get more passives off, which means you use fewer abilities to do damage. It also means you can waveclear without using Q, thanks to the increased damage and the AoE on your passive.
I didn't say not to build Zhonya's, I said to only built it first against a Zed, and second only if the enemy laner is AD. If neither of those cases are true, you don't need it until mid-late game teamfights, which means it's safe to build as a third or fourth item, especially now that the AP has been nerfed.
Re: Abyssal's passive range: It's 700 units. For reference, Diana's maximum Q range is 830. The only situation where I wouldn't buy one is if the enemy team had no high-damage casters.
Lich Bane also synergizes well: it boosts your initial burst, helps you catch, gives you more mana to work with, and also helps you take down towers once you push the enemy laner out of lane.
Re: Prepping passive making opponents wary: And? Either they're dumb and let you get the extra burst, or they back off and let you zone them. That's one thing that I notice a lot of people not understanding: you don't need to kill the enemy laner to make them useless. If they're scared of you, ward the river and stay in an aggressive trading position with your passive active as much as possible. If they try to CS (and aren't Xerath or something who can do it from half the map away), walk farther to keep them back. You probably won't be able to keep them out of exp range in mid, but you can prevent them from farming.
Annie is an example of why Abyssal is good, and of the value of knowing when to play safe. That's the sort of situation where you don't try to kill at all unless they're stupid, you just farm as best you can and ask for ganks, then roam for kills once you have an item or two. If you really want to try for kills, stay safe until 6 and wait for opportunities. With Annie, that's when she's been holding a stun and has just used it to farm. Her ratios and base damage are higher than yours, so you can only fight when some of her abilities are on CD and she doesn't have stun (preferably also only after you've built MR).
Wanting to build straight AP on Diana is silly for a couple reasons. Champions like Zed and LB can build straight damage because they have escapes. Xerath and Vel'Koz can do it because they've got really long ranges and plenty of CC. Veigar can do it because his sole contribution is landing a good stun and deleting one enemy. You can't; you have mobility, but it's entirely dedicated to offense. Once a Diana goes in, either she kills the enemy or she dies. That's why you need some defensive stats, that's why you need the extra damage from Nashor's -> passive and Lich Bane. If you build straight AP and go in during a teamfight, you're going to get off one Q+R+W and die. Do you also forgo scaling MR blues because you want more AP? No, of course not.
Diana's sort of unusual, in that while she's technically an assassin, a lot of her damage isn't frontloaded. You need time to get your passive out, time for your W's spheres to all rotate around, time to land a second R. If you're squishy, you don't get that time, which means you're losing out on a lot of damage. It doesn't matter how much AP you have if you only hit the enemy with Q+R+one W sphere. Nashor's helps with the passive while also giving other valuable stats. Lich Bane gives some extra initial burst and followup burst after your second R as well. Abyssal + Zhonya's gives you enough resistances to not die instantly plus a chance to refresh CDs after your second round of burst after your second R.
If you're just going to build AP, at least do your team a favor and rush a Deathcap instead of Luden's.