Ridiculously slow MacBook Pro 15'' 2011 with low battery and El Capitan Problem solved: see "Update 2" section.

I have a MacBook Pro 15'', A1286, Early 2011 which was running OS X Lion and became extremely slow few weeks after an upgrade to El Capitan.
The upgrade itself took very long. After the upgrade, the laptop was running smoothly during about 1 week. 

Then, performance suddenly dropped and all menus now take a while to open.
The mouse cursor is very jerky. The one of the touchpad is also jerky, altough to a slightly less extent. Practically, it is impossible to use the laptop.

The battery is no longer charging and the time (date/hour) is no longer remembered, so that I have to reset it after each start of the OS.

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Specifications : Processor i7 @ 2,3 GHz with 4 GB RAM
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Update 1


Upgraded to 8 GB RAM. 
Shrinked OS partition. 
Cloned to SSD.  
Checked SMART values.  
Checked file system.


Problem not solved.

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Update 2

Short version:

The culprit is the low battery
After thorough tests, I came to the conclusion that the jerky cursor movements and the MacBook coming slower and slower was caused by the battery with "replace now" status.



Long version or how the problem has been identified. 

I first followed this procedure in the comments of a video showing how to fix battery issue on a MacBook: 


Power off computer
Remove battery 
Disconnect magsafe 
Connect magsafe 
Connect batte
Power on


This had the positive effect that afterwards, the date/time were remembered, although the battery was in very poor ("replace now") condition.

Then, I reset SMC and NVRAM, following the procedures linked by users Nimesh Neema and Amey. The problem remained. The MacBook was fast during the first seconds and then went slower and slower. This happened even when using softwares from Apple only and not from third-parties.

Lastly, I tested the MacBook with the battery from a 13'' inch (A1278) MacBook Pro. Both batteries operates at the same voltage (although thus of the 15'' has higher capacity). The battery connector is not a the same location, but is same, so that I could insert it and hold the battery with my hand.
This battery was in better condition (with "must be checked" status, but didn't had yet the "replace now" status. This totally solved the problem. The MacBook remained fast and the cursor movements, from the touchpad or mouse were still smooth after several minutes.



Conclusion: If the battery has the "replace now" status, some artifacts totally paralyze the MacBook Pro, although it appears working correctly during the first seconds. This is a purely hardware problem and system reinstall or upgrade is NOT necessary.
Perform Apple Hardware Test to check for any hardware failures. To do so, power up your Mac, then immediately press and hold the D key. Keep holding the key until you see the Apple Hardware Test icon. Thereafter follow the steps as mentioned in the support article, How to use Apple Hardware Test on your Mac.
Reset SMC and NVRAM.
If the hardware test passes, try reinstalling OS X El Capitan. If the problem persists, fallback to Mac OS X Lion.