I'm aiming on winning team-fights, teamfights happen at around the same time you finish those first 3 items too.Moreover, Runaan makes teemo a pushing machine, massive farming and pushing is quite gold efficient. You also get more minions on your side in 1vs1 fights, lol. But yes, Blade of the Ruined King better for 1vs1, but I'm looking beyond. GIve me some time on the Attack speed Runes, it seems to help me poke and farm at the same time, but I'll give other runes a few trys.

Attack speed doesn't work that well on Teemo. You are better at building off full bruiser or AP.

Runaan's Hurricane really isn't gold-efficient item unless you can actually hit your bolts. If you want to 1 v 1 someone, get Blade of the Ruined King.

pretty horrible early-mid game items?
For most champions, these wouldn't be enough for damage, but Teemo has TOXIC SHOT. HE needs the attack speed as I mentioned in the Items section, perhaps i wasn't clear? More attkck speed means more Toxic shot stacks= more damage. Once you get Runann, you can 1vs1 most top champions especially the non-burst types. Or even just a curve bow. Imagine teemo has a free Pick Axe, that is what Toxic shot is like. Once i get these 3 core items, Team fights happen and I often end up win the team fights, so give it a try..
You have no alternatives
These three items, Runaan and Fzn Mallet make him a bruiser, he needs it for the combo as mentioned...
The way to adapt for teemo is to build what you need first within those 3 core items + vampire scepter if needed. I can lane against any top champion. For burst, rush some health, for bruiser that trade, get attack speed for damage, and for pokes, get vampire scepter. Lots of alternatives.
You can get other boots too if u want, but berserkers are best early game if you want to kill the enemy. But if you just wanna defend, go ahead and get other boots.
Runes:
Same reason as attack speed items- TOXIC Shot. Faster atk speed also means easier farm and faster hit and runs. You can try damage or hybrid marks too, if you find it hard to last hit; I haven't done extensive testing on runes, so I'll try them. Thx for reply! I'm sry if I didn't make myself clear on this in items section already; just try it out, it works for teemo.

Berserker's Greaves, Runaan's Hurricane and Frozen Mallet are pretty horrible early-mid game items

You have no alternatives

You can't use your skill points like that

Mark of Attack Speed are practically useless as you aren't jungler

you should explain, well, everything and go more in-depth with the explanations