You can certainly have nice spirals and spins. You may not be naturally flexible enough to ever achieve the super high spirals you see on TV. But if you stretch regularly and don't have any hip or knee problems, you can get a very nice spiral above hip level.
Spins seem to be a lot harder for adults than they are for kids (well, everything is harder for us) but again,
with practice, they can be beautiful.
I know a lot of people who started skating as adults and are now working on axels, including me. I don't know anyone who's landed one when they didn't begin skating until adulthood, though I'm sure someone here has. But most of those adults working on them have explanations like, "Well, I was close, and then I got pregnant..." and "Work got really busy so I couldn't skate." So landing an axel isn't impossible but it does take consistent focus that a lot of adults just aren't able to have.
But the axel is just a jump! You can be a beautiful skater without one and a sloppy skater with one.