Like jelger my experience in the area is driving. Nearly all of it in September(s) and October(s) when I've driven a good portion of the Black Sea coast of Turkey.
1) Yes, you can expect quite a bit of wet weather along the coast. Hopa to Samsun weather is normally grey and misty, especially in the mornings. As the year progresses SERIOUS rain events become more and more possible.
2) The coastal route between Hopa and Samsun is mainly flat BUT there are MANY tunnels through roughly N-S offshoots of the E-W mountain range. If bicycles are not allowed in the tunnels (VERY possible) you face some serious climbs and long diversions just to make a small amount of "real" progress along your route. West of Samsun, the coast becomes more and more mountainous to become essentially impossible for the 150 miles or so between Gerze and Ereğli.
3) You can take a reasonable trip through the coastal mountain range by following the E95 from Samsun to the E80 that leads to Istanbul.
4) Turkey is NOT bicycle friendly! Utterly zero consideration for bicycles in the design of ANYTHING. Why? Because they are neither a practical nor reasonable means of transportation (or even enjoyment for kids) in the vast majority of the country!