1) To troubleshoot the mail issues, I would start by reducing to just one account and making sure you have everything working. For gmail, I prefer using the gmail app rather than the main mail app (which is samsung-ified" on the S4). For your yahoo accounts you may need to use the yahoo mail app as yahoo doesn't have IMAP which is what keeps your read status in sync. Exchange should work with the mail app with bi-directional sync though.

2) Photos can be placed in folders either on the device itself or just by connecting to your pc and doing it manually via windows explorer. The gallery app has some settings to auto-sort too so might be worth looking at that.

3) Not sure why your appointments wouldn't sync - although I'm not familiar with SPlanner

4) This might just be a case of too much data to import - is there any way to do this in chunks?