The bike is almost certainly too big, but not outside the range where a competent fitter should be able to make it comfortable for you. You may sacrifice a little power and the handling will not be as good as it could be, but unless you're racing you probably won't notice (until you eventually switch to a smaller bike).

The foot pain is atypical but it could be lots of things. Shoes that are too flexible, incorrect arch support, toe-clips not putting the pedal spindle in the right spot under your foot, etc. Your fitter should be able to narrow it down.

Hand pain is common and is often a result of saddle fore/aft position. There are people who will debate to the death which way to move the saddle to take weight off the hands, but your fitter will be able to experiment to find what works for you. I assume you're riding with hands on the brake hoods, so bar position and angle and shifter position will also factor in. Saddle angle will also contribute. Beginners have a huge tendency to set the saddle "nose down". This puts constant pressure on your arms hands as they try to hold you back from sliding forward. Even as little as 1 degree down can be way too much.

Lower back pain is also common. Often it's matter of not having enough flexibility or simply riding in the wrong position. You shouldn't be "arching" your back when you ride. You lean forward by rotating the pelvis. Tight glutes and hamstrings can make this difficult. The goal is for your back to be a flat as possible when you're in riding position. If functional limitations prevent this, then the solution is often to raise the handlebars and begin a stretching program. As you gain flexibility, you can adjust your position back down. Back pain is also sometimes a result of underdeveloped core muscles and will simply go away as you get stronger. Your fitter should be able to distinguish between the two.

Good luck. Bike fitting isn't a cookbook process and not everyone calling themselves a fitter is actually qualified. Lots of stuff interacts and knowing what to change to fix one problem without aggravating or creating another isn't easily taught. If the shop's fitter isn't skilled enough, find another. If that's cost prohibitive, then hit up the Bike Fit forum and folks will try to help.