Depends on the theme. With some themes everything is retained. With some you lose the text widgets, so it's best to be cautious about that (you don't have to retype them, just copy their content and paste it in a simple text editor). When you switch to a theme with a different number of columns, changes are bound to happen.
Rearranging the widget order is very easy: you just drag them into new positions.
Page hierarchies are not theme-dependent. The presentation might be different (e.g. bullets or no bullets, indentation, etc.).
And so that you won't be unnecessarily alarmed, if you're not familiar with that: with some themes you'll initially get their own set of widgets and/or a "you're using 0 widgets" note. Click "Save Changes" or add any widget and enjoy the result.