As a regular Poppy player and self-styled "Poppy fan", I have to say the only thing that keeps her out of high-level play is the all-powerful Meta. While an art update is sorely needed, there's not much room for change in her kit without a giant rework (I'm talking gutting and starting over). A mana-taxcut would mean she could snowball earlier, allowing her to hyper-carry as early as the 25-minute mark, and that's just no fun.
Speaking of "no fun", her two unfairly-powerful abilities (her passive and her ult) are enough to have RIOT on-edge about even mentioning Poppy's name, let alone changing her kit. Last I checked, I read somewhere (don't ask me where, probably on Reddit) that RIOT's position on the matter was that she's due for a rework, but they're afraid to drastically change something that works as of now. Not to say that she sees a ton of play, but that those who know how to play her get the advantage of an ult with no method of counter-play, and a passive that more-or-less keeps you safe from any and all assassination-attempts.
If you're having early-game trouble, try playing with health-regen quints and taking Summoner Heal and Summoner Barrier. You be surprised how often your lane-opponent will misjudge your survivability under your tower and give up easy kills. As for laning/last-hitting, I like starting W and making sure to keep hitting things to keep the 10 stacks up. Yes, her W has a crazy-high mana-cost, but it does pay for itself if you can just keep on hitting things (and trust me, those runes/spells will keep you alive until your jungler shows up).
Keep trying, and remember there's always room to think outside the meta!