With a Name Change:
For all servers, you will get your donator rank and any other items purchased on the store. A name change also does not clear your previous punishments, such as mutes and bans.
Skywars: You keep everything
Kitpvp: You keep everything
On Parkour, Skyblock, Prison, Creative, you will lose money, gems and echest items unless given to a trusted player to be returned after your name change. More details below.
Parkour: You lose your parkour rank, for example, Expert. You lose items in your inventory. These could be given to a friend before changing names.
Skyblock: You lose mcmmo. Your island will be returned to you, but you must be online with someone who can return it (typically me). Your chests on your island will be returned to you.
Prison: Your prison rank and plots will be restored.
Creative: Your plots and grade will be returned to you.
As servers are updated, they will all become compatible with name changes. But that is a work in progress. If I missed covering anything, please let me know and I can edit this information. This is what I remember off the top of my head.