I recently downloaded this game to my iPhone 6 plus, and I played for several hours and gained at least 6 levels for Merisiel and Kyra through Story and Quest modes. When I logged back in this morning, all the levels and character progression was reset to zero. I had finished Perils of the Lost Coast and purchased Burnt Offerings, and the progress there remains the same as does the gold that I had earned. Is this normal? Do I need to start character levels over every time I re-log in? This seems very counter productive and I can't see how to progress through the game like this and I surely don't want to spend any real money on this game should this be the case.

In party management if you look at the top left corner you see the options New and Experienced. If you toggle Experienced your leveled characters should show up.

I did try that, but it still didn't change it. And today when I played, I got gold for finishing areas, but no XP at all. Do you only gain XP in certain play modes, like 'Quest'?

Only for the first play or legendary difficulty all times.

In Story mode you get Gold the first time you successfully complete a scenario on Normal and Heroic difficulty. You get a Gold reward on Legendary every time you succeed. Character advancement in Story mode is only via Scenario Rewards

 

In Quest mode, you get XP every time you finish a Scenario - you get a bonus if you succeed and you get more XP for Heroic and Legendary difficulties. Character advancement is by accumulated XP. You also get Gold for Quest mode but only as a bonus for success (though there are small rewards during the scenario for defeating Banes, you get this regardless of success or failure)

 

Or this was the case up to AD5, I haven't played since around November, I guess something may have changed with a more recent patch.