A recent thread on BF discussed users' experiences with Microshift components. The overall impression I gained from reading it is that the small upshift lever on Microshift shifters is not truly accessible for shifting in the drops.
Microshift operates with the same actuation ratios as Shimano, so your derailleurs are fine. Normally, it's nice to install new housing with the new cables, but your bike is fairly new, and your existing housing is likely in fine condition. I'd probably replace the housing loop at the rear dérailleur because it tends to wear more quickly. You could ask a shop to cut a section to correct length if you don't have a cutter.
Installing the shifters will require tuning the derailleurs. You'll also have to either install new tape or rewrap your existing tape if it's in good condition. Both of these tasks can be frustrating for first timers.