Battery capacity wear-down, and its relationship to charging practices
Firstly, it appears that discharge cycle has been misunderstood. It covers both charge and discharge and is a metric to cover battery life. (See this for more details Android definition of charge cycle?).

Ideal charge / discharge limits in the question Ideal charging / discharging percentage for maximum battery life? discusses how to increase the number of such cycles and thereby life of battery



  Does battery discharge cycle have any relationship with battery capacity?


Yes

The more you use the battery , more the number of discharge cycles ( which covers both charging and discharging as explained earlier) and drop in battery capacity as illustrated in this figure, which shows capacity drop as the number of cycles increase for Li Ion batteries of different capacities



Source


  What factors could affect battery capacity wear-down?



Cycling or the amount of use in terms of charge cycles covered above
Elevated temperature : 


Physically exposing the device to high temperatures ( like on a dashboard of car on a sunny day) or 
Increase in internal temperature, by 
keeping a cell at a high charge voltage (hence the reference to ideal voltages, which doesn't permit build up of temperature )




  Exposing the battery to high temperature and dwelling in a full state-of-charge for an extended time can be more stressful than cycling



Aging: This source at the end talks of the chemical reasons that cause aging. Without getting into the chemistry of it, what it means is that the capacity of battery starts decreasing from the day it is manufactured, represented by the active zone in figure




Additional Information


How to Prolong Lithium-based Batteries
Charging tag wiki, especially the General and Maintenance section for good practices
Some OEMs at times provide a battery of higher capacity than rated - this would partially offset aging effects. See  Honor 6 with 3100mAh battery against 3000 mAh rating


Edit  Accubattery Pro gives you an indication of estimated Vs actual capacity as shown below. I have no idea how accurate it is but surely can be indicative