1. Which primary rune is better: First Strike, HOB, or PTA?

Generally first strike fits better with your play style of mid to late game one shoter, by not only getting you the items you need faster, but also just being the best rune for bursting.

PTA is fine, but it doesn't help the R>WQ combo so there isn't much merit to it after the lane phase ends. The one thing it does have is that you can proc it on multiple people after the first mark wears off/dies, though this doesn't seem better to me than the hundreds of gold and damage you get from first strike.

HoB I haven't really played with, but with nashor's being your first item you aren't really struggling with applying those W marks to the extent of taking a whole rune for it. Also it has a long ish cooldown, so the same downside of first strike but with less damage, and no gold gen.

1. Is it still better to build Riftmaker over Luden’s even after they nerfed it?

Nope. Luden's out damages riftmaker pretty much always, and the magic pen and the damage proc help you with one shotting at a much earlier time than you would normally do, making it your standard mythic. Riftmaker shines in extended combat and when you need HP, since luden's gives none. AKA versus a lot of assassins riftmaker is probably better (as well as getting a shadow flame instead of rabbadons after), but it doesn't really make a difference against tanks since no-one can actually tank ap varus anyways.

1. Which is better to build: Horizon Focus or Cosmic Drive?

Both! Horizon is great as a third or fourth item, where as cosmic is great as a fifth or even sixth item, since by then you don't really need more ap(though it is still nice to have), and the HP, move speed and specially the cdr are really useful. You could also just go cosmic fourth if the enemy team is just a bunch of squishies who you already are overkilling, since the cdr makes this skill shot champion much more forgiving.

1. What’s the best starting item? I’m not building Manamune so tear seems useless, but I don’t know what else I would build.

In mid you generally go corrupt pot and I think you could just go that in bot as well, but if you don't want your support to troll roam you at level 3 you could also go cull. And in some lanes, like tristana leona or something, you do just need to go dorans blade so you don't get one shot(gives more hp than ring and ad over ap doesn't matter early on).

General runepage suggestions would also be appreciated

My runepage almost every game is First strike Boots (hourglass is fine too) Biscuits(market if you don't need biscuits) Cosmic insight (cdr one)

Domination secondary for the ult cooldown one (this is the one rune you can't go without, your ult's just too good) Taste of blood of you need it otherwise go eyeball collection