Since I do not know the characters yet, although I see the preview when I hit the "?" button, it disappears as I start writing. Maybe it's my short term memory, but I could use the ability to trace it a few times before it goes away :)
Any way to do that?

you could try pressing S when it goes away (and repeat)

I know this doesn't really answer you about keeping the character onscreen, but in this type of situation where I just really don't know the character I keep a notebook by the computer and draw it there a few times. Then you can erase and rewrite a few times on Skritter too, if that helps.

No, there isn't. I remember asking Nick the same thing when I first started using Skritter. He said they'd experimented with this, but found having the character disappear was much more effective.

Maybe you could write the characters on paper a few times while you keep them on screen, until you get more used to them?

Starting with the radicals list might be helpful, 'cos later you'll be able to break more complicated characters down in your head, component-by-component rather than stroke-by-stroke.

Could always copy and past the sentence into a text editor

Erase and rewrite it a couple of times if you need to.

As Lyons mentioned, we found it's just not very effective to write over the character; it's best to immediately push your brain to remember as much as it can on its own. Press the show button as many times as you need while writing the character, but only as often as you need. Then erase what you wrote and write it again, trying to use the show button less than before. Repeat until you can write the character the whole way through without using the show button.

I would like to be able to preview a section before starting to write. I know there are lists you can look at, but I would like to see the characters in the larger size of the study section.

Byki, which I use, does this.