New surveys use the theme Fruity TwentyThree. Its theme options set the logo, the font and the colors of your survey, without code. You find them in the editor under Presentation, with a preview of your survey on the right.
Every change saves by itself. Saved at in the top bar shows the time of the last save.
Tip: To set a look for all surveys of your survey site, use the theme options on the Themes page: under Configuration > Advanced > Themes, click ... next to the theme, then Theme options. Surveys inherit them unless they set their own.
Yes, in your own theme. See Create your own theme in the theme editor.
Yes, on the Expert, Business and Corporate plans. See How do I remove the LimeSurvey branding?