Is poppy viable in 1800+ elos? It seems as though her ult and movespeed could completely dominate late game team fights, and she is a fairly finesse-oriented champ what with her e's, but I rarely if ever see her played at higher elos. Is she just not viable? Laning phase too weak? What is it that makes her so under-represented?

Now I'm not 1800 but I've done a lot of reading on Poppy and have run her a few times and what I find as her glaring weakness is her laning phase. She is harassed very easily and runs out of mana VERY quickly if she retaliates. She just has to farm and stick with it. The more common top lane champs can force her out of lane very easily. Yes if you manage to farm well enough and not die in the laning phase you will transition to become the powerhouse she can be endgame but you're extremely vulnerable early on.

Farming meta > champs who have issues farming

no ppl dont play her in higher elos and pro dont really player her ether she to easy to shut down she need a rework her q and r are great but her charge is easy to stop and even if u dont stop it just dont be bear a wall and u wont get stun her w doesnt buff her enough to make her that big a threat ether. and before ppl bash what im saying play her rank and top lane solo before u say anything

I know of ONE guy who played her at high ELO succesfully. So it's certainly possible, but not exactly easy.

She counters people like Garen/Riven/GP top and has one of the strongest mid-late games in the League.

Short answer: no.

Long answer: There's one guy who mains her at 2200 ELO or so, but he's probably pushing the utmost limits of Poppy's potential and likely could have done even better with a different champ. On the other hand, she does extremely well in Dominion where there's no laning phase to hold her back and her crippling mana problems are partially overcome by healthpacks and global mana regen buff. Even then she's less of a roflstomper and more just a generally strong pick. Strong ranged CC can help shut her down, and there's a few who can out-duel her, but she'll serve most Dominion teams well and she's certainly competitive there. Not so much in Summoner's Rift though.