Originally Posted by cpach

Literally the only shifters that will work correctly are the 4700 flatbar shifters because 4700 uses a different cable pull ratio than other Shimano 10sp, so yeah, definitely those. You will need short pull brake levers if using those same brakes--or take this as an opportunity to upgrade to hydraulic. If hydraulic, the absolute cheapest Shimano brakes work just fine, the next step up is at the Deore level, beyond that it's mostly not a big deal.

The bars you mentioned sound reasonable. Flat bar/backswept bar road bikes aren't my personal cup of tea so I don't have very personal recommendations regarding them.