I have no doubt that is oak. The cross/diagonal grain cut looks really nice. It can be challenging to work with when cut that way, so nice going there. The prism finish looks pretty good with it, too. I
highly encourage you to look through some of the many thousands of pictures in the gallery to see what appeals to you and even start some ideas brewing. Some of the designs are pretty mind boggling.
As far a shape goes, my first pen looked like a snake that had swallowed an egg. One thing to keep in mind is that what looks like a "slight" curve on the lathe/mandrel is much more pronounced once it's on the finished pen. There are a lot of penmakers that carve their pens like you did here, so do what works for you. Unless you decide to sell some, then do what makes the money. That being said, my personal preference is also something thinner, or with a subtle, gently flowing curve. The last couple slimlines I made were completely straight and I wasn't happy with that look, either. Still experimenting and learning...
Mostly, just keep going and see where it takes you. Besides the poorhouse, that is.