Lets start with the easy stuff. Take the battery to some place and have it tested. Newer cars are really picky about the battery's and do strange things when they are close to failing, but not yet dead. Checks out? Get a pressure gauge and check the fuel pressure. It's OK. Rats, its electrical. Look for trouble codes. None? Rats again. I say goodbye to my money.

The last time something like this happened to me it was the battery, and it didn't give any sighs of being weak.

Could you have 1 bad plug or coil pack? Kinda sounds like a cylinder is not right with the rough idle and bad fuel economy.

Les