Right now, it looks as if the only trail that remains closed is the upper part of Sugarland Mountain Trail from Huskey Gap back up to the AT, so you shouldn't be affected that much.
Otherwise, there are a tremendous number of options with Gatlinburg as your base. If you're looking at high country, I think you might consider Andrews Bald--it's a couple of miles down the Forney Ridge Trail from the Clingman's Dome parking lot...a bit of elevation drop and gain, but probably doable for your kids. You can also find a lot of nice mid-elevation hikes in the Elkmont area--Little River Trail is basically a pretty flat gravel road, and you can make a nice, easy loop by connecting with Cucumber Gap Trail.
As for "cautions," I'd mainly just give myself time to get from here to there--there will be a lot of tourists in the area, especially at the beginning of wildflower season. (Although some sections of trail can be closed for bear activity as well.)
Here's a link to the NPS site--there's also a Twitter feed on road conditions, too.
At any rate, I know the Park pretty well, so feel free to ask as your travel plans firm up.