What could be the source of my Samsung UN55ES6500 LED TV's picture going out? I still have a picture, but the color is weird and it leaves a trace as an object moves across the screen.

Fault at its main digital board. Short circuited component/s at it. Contact any service technician; or the authorized service center. If you wish to get some details; check the site linked here. Surf the site with patience. Pull up older posts.
http://electronicshelponline.blogspot.com/
In most cases, the main board might have to be replaced. If so, a software [firmware] update too might be necessary.
Before going further; try a hard reset first. If it does not help you, replace the main digital board. There are two methods to reset TVs. The first is simple to do; a Hard Reset. The other is somewhat skilled; is a Factory Reset (System Reset). To make it a hard reset, just unplug the TV from AC mains wall socket, and re-plug it back after 15 minutes or so. If you do this by overnight is better. This procedure will make it a hard reset. If you want to make it a system reset [Factory Reset]; you have to enter its service mode option, and select the factory reset option. If you wish to get some details; visit the site linked here.

Setup menu - WiFi - LAN - Network if this does not work try going to web browser to get an error message.
Samsung 3D LED Smart TV Setup menu WiFi LAN Network

This is probably very simple in the fact that your button is too sensitive. I have run into this with remotes a few times. Replace the remote and the problem should go away.

You'll need to replace the main tuner board. You can get it onlie from:
TV Parts Universal TV Stands DLP Chips ShopJimmy It's pretty easy to do yourself.

No repair to a cracked screen; other than replacement.

Same exact thing going on with ours !!! Same model !

If you wish to get some details; check the site linked here. Pull up older posts. Surf the site with patience. http://electronicshelponline.blogspot.com/
Line, faults & possible causes to LCD screen are given with illustrations. You can get an idea about the fault to an extent.

Yes it does

Back-light problem. Check and replace damaged component/s at its back-light inverter board, or replace this board as card basis. If you wish to get some details; check the site linked here. Surf the site with patience. Pull up older posts. It will be best to replace the inverter board as card basis.
http://electronicshelponline.blogspot.com/