1. Should I continue using the synthetic oil? Previous owner was high on it, local dealer is fence sitting.
There are benefits to using synthetic oil. 100% synthetic oil (not blend) will result in less engine wear down the road and better overall lubricating properties. The big thing with running synthetics is that they can go at least twice as long as regular petroleum based oil in between oil changes. However, some that run synthetics still change every 3-4K anyway just to be safe, but others (including myself) run 6-7K on synthetic oil no problem.
2. Should I get the bubbling paint fixed on the lift gate? Whine to Ford? Bite the bullet?
If it's bad I'd get it fixed. The warranty is over with so whining to Ford will solve nothing except for wasting your time and effort while the dealer laughs in your face. It's easy to do yourself; there are tons of online guides for body work. If it's rusted through however it'd be advisable to take it to a reputable shop.
3. Which floor mats are best? Carpet? Molded plastic?
I find carpet mats will only last a couple years of winter before they look bad, and 3-4 years before they look terrible. Unless you drive in bare or stocking feet then get carpet, if you drive with footwear on like the majority of ppl go for moulded plastic. WeatherTech Automotive Accessories - Car Mats, Floor Mats, Cargo Liners, Side Window Deflectors and More
should make some that are custom fit and will last several years.
4. The hub caps are corroded where painted. Don't worry about it? Strip and repaint? Replace if it drives me nuts?
Hub caps can be replaced. Ebay would be a good start as some can be found for cheap ($0.99 opening bid). It beats buying new at the dealer. You could repaint, or find a local chrome shop for a bit of a customized look depending on the type of wheels you have.
5. It's a huge vehicle with no internal storage (jumper cables, tow strap, etc.) What do you all do for more places to keep important accessories? Plastic boxes?
Plastic is right. I used old plastic boxes in my old trucks but they also held tools as well as a spare serp. belt and various accessory items. I would have thought that you'd have some sort of underfloor storage? Hmm.