Check the ac/dc adaptor with a multi meter it should match the writing on the unit. they can run hot and burn out it might be possible to have 2 broken units.
disabling or enabling devices that use power can be used to diagnose a battery problem e.g. blue tooth fire-wire thunderbolt SD CD and DVD drives
Disabling OR enabling as many as possible and changing the settings to low power or high performance should make it last longer or less long. either of these situations confirm a battery problem in a case of less or more charge. a logic board problem would make no difference if you get my drift. The idea is a normal battery would not suffer much impact like this in short term use but a dodgy battery would be very sensitive. Its not really relevant if you can't boot because of no charge.
I've had battery problems before these are my steps I know you have probably tried already but if your desperate it might be worth trying again.
take the back off your mac and check to see all the ribbons are undamaged also check to make sure the the ribbon connectors are not cracked twisted or broken its a common problem when unplugging the battery to damage the connector they are very delicate and require a very light touch.. When unplugging them you cant just flip them out they need even pressure from each side and the middle and gentle levering very tiny amounts at a time if you are in the habit of flipping them out you may have inadvertently damaged them, if you think you may have been careless with the connectors you can get replacements inexpensively and this is a common issue on the mac.
i would disconnect the battery from the logic board and leave it for 15 mins then be very careful to firmly plug it back in using even pressure so all of it is plugged in evenly, be very careful. don't connect the mag. (trying to discharge the mac battery while the mains is plugged in won't discharge the battery). Then close your mac back up and screw it back together. before you plug it in the mag hold down the power button to 15 seconds or longer, something should happen a beep, a flashing light so just wait. if after a while nothing happens plug in the mag. But just wait for something to happen, mac batteries have an issue where they can get all screwed up and need discharging and this is how I would fix it.
the main reason for a battery needing this is usually due to hardware changes (for example recently I'm betting bet you have made some hardware changes like an SSD or just better memory), the mac will work fine for a while after a hardware change then need a negative discharge, its a one time deal and due to a new power setup from voltage changes in the hardware. Replacing the battery won't change the situation as the system still needs to reconfigure its new voltage settings. Don't try to reset the battery with the mains pugged in its all done without any external power, i think its a lot to do with the boot up process and probably unique to MBP.
---edit i edited this to make it more informative and corrected spellings---
Is it correct that the charging with value = 5.5w, shouldn't it be c.
60w?
This is expected behavior.
The 60W rating on the power adapter is the maximum amount of power that the adapter will supply. Note that it's not the amount that will be "pushed," but the amount that can be drawn from it.
Slow Charging Speed
... for my 15" late 2012 MBP (non retina)...I'm currently charging on a 60W original Apple charger (as I
understand it, the 85W is optimal, but not necessary).
This is an incorrect assumption on your part. The 2012 15" MacBook Pro requires the 85W charger - see Apple: Find the right power adapter and cable for your Mac notebook
This is why you're seeing the very long charging times - your MacBook Pro is "asking" for more than your adapter can deliver and since the adapter can't supply any more, it lowers the power to the battery charger in favor of the compute requirements.
Your MacBook has two operations going on: powering the system and charging the battery. In this scenario, priority is given to powering the system. If you were to shut it down, the priority would naturally go to charging the battery.
Seeing only 5 to 6W of power being allocated to a running computer is perfectly fine (for the 60W charger); the power is being divided to account for your current use. Use the correct charger and it will operate and charge correctly.
Further Reading
There's additional questions/answers on this very topic. Please see the following links for further reading:
TL;DR
You're using the wrong charger. Get the 85W charging adapter and your charge times will decrease to normal levels.
Best Answer
Reset the SMC
I would start by resetting the SMC on your MBP. Follow these steps:
Once you've reset the SMC use your MBP as usual. If you still feel it's shutting down too soon or perhaps not charging as it normally would, then reset the SMC again but with the following steps instead:
Also, macOS provides a great deal of information about your power usage. Read on to see how to access this.
Checking your charging status within macOS
Follow these steps:
Go to Apple > About This Mac
Click on the System Report... button
In the left pane under the Hardware heading, select Power (it's about 2/3 of the way down)
On the right-hand side, scroll down to view the AC Charger Information