It sounds to me like you're looking for a solid All-Mountain ski. Something with a little bit of early rise in the tip, and a decently wide platform to stand on. As far as twin tip vs. flat tail, that's really all up to you. A lot of people might say you don't want a twin tip because they're meant for freestyle tricks, but in reality a good twin tip can hold its own anywhere on the mountain. Chances are you won't want any metal in your skis yet, so really the difference between some twin tips and flat tails skis is pretty minmal.
Now, let's get into some suggestions:
1. 2013 Bizzard Bushwacker- These skis are a solid 88mm underfoot, have a bit of early rise, and a full wood core. These would be ideal for you right now as they're narrow enough to be able to handle, yet wide enough for you to go off trail when you have to. The full wood core provides solid support for your skill level, and the exclusion of metal ensures that they won't be too stiff for you. I'd recommend the 166cm length for you, though the 159cm would be a bit quicker for you to pick up skiing with.
2. 2014 Rossignol Experience 88- These skis are essentially Rossignol's Bushwacker. Next year's model will mark the third year of the Experience series, and they get rave reviews year after year. The waist on these is an 88, and they're also a full wood core. The big difference between these skis and the Bushwacker, is the Experience has a lower turn radius (quicker, easier turns), and the Bushwacker uses Blizzard's FlipCore (easily the most stable rocker construction on the market). Plus, this is the 2014 model, so you'll likely be the only one riding these at your mountain until next year. I'd recommend the 162cm length in this ski.
3. 2013 Atomic Alibi- This is an example of a Twin Tip ski that can handle skiing the whole mountain just fine. These skis are a bit wider in the waist, coming in at 95mm, so that's definitely something to consider. That width might be a little bit too much at first, but it seems like something you could work into pretty quickly, given you've already skied a couple of double black diamonds. They also feature Atomic's All Terrain Rocker, which means they've got early rise rocker in the tip and tails, with camber underfoot. These would be a great choice if you think you'll be checking out powder stashes on your skis. In that sense, the wider waist gives these the upper hand compared to the Bushwacker and the Experience 88. I'd recommend the 168cm here, but you'll have to be ready to step up to the plate, or you'll find these skis a bit cumbersome.
4. 2013 Volkl Bridge- This is another All Mountain Twin Tip that would satisfy your needs. It's very similar to the Atomic Alibi, except the rocker profile is a bit different. On these skis, you'll be getting Volkl's full rocker, which means these skis have a continuous, low rocker profile from tip to tail. There's no camber underfoot, so they won't be able to grip the ice or hardpack on firm days quite as well as the other choices. The benefit to this ski is that it's super easy to turn, and it'll handle some of the soft snow when you're ready to search that out. Plus, it satisfies your attraction to a twin tip ski. I'd recommend a 163cm length in this ski.
Obviously there are a lot of other choices out there that could work for you. Another ski worth mentioning would be the Dynastar Cham 87. Feel free to ask any questions, and good luck with the search!
Matt w/ Skiessentials.com