This is actually hard to give any answer without knowing what exactly inside of your chargers.
But I'd say that any charger should have a diode on its positive output (cathode to output) so it should be safe. However it is pointless to connect two chargers simultaneously for the same (diode) reason (the charger giving lower voltage will not open its output diode and there will be no current at all).
I'd suggest to pick only one charger which is more powerful (basing on its description or on the chargers amperemeter). This is safe and 99% fastest way.
I managed to obtain 6 x 18650 Batteries from an old laptop.
This is your first problem. Those old batteries are probably tired and will struggle to supply the required current. Individual cells may have different internal resistances and capacities, so balancing is advised.
Solution A - Use only a 1S3P (or more in parallel) Pack instead and
use a TP4056-based USB 5V Charger.
Bad idea. The battery will charge very slowly, and the booster will waste power. The pack and wiring will have to handle 14A+ discharge current.
Solution B (BMS and '12.6V' charger)
If the BMS includes balancing then it should work, provided the '12.6V' charger is designed for 3.7V Lithium cells. Without balancing, some cells could reach peak voltage before others and then the BMS would terminate the charge early, resulting in a partially charged, out of balance battery.
The BMS won't cut on discharge until at least one cell has dropped to a dangerously low voltage. After a few cycles the cells will start dying. To protect the battery you should install an alarm or cutoff that doesn't let any cell go below 3.2V.
Solution C - Individually Protect Each CELL with a 1S BMS, AND use a
3S BMS
Overkill, but perhaps (depending on the balancers) not enough! Many balancers work on the principle of bypassing charging current when the cell reaches peak voltage (4.2V). The problem with this method is that if the balancer can't bypass all the current then the cell will continue to be overcharged (until the protection circuit kicks in).
Solution D - The Proper Balanced Method , which would need a use a of
bulky balance charger
Again, how well this will work depends on the particular charger. Some contain 3 isolated circuits that charge each cell individually. This is the most reliable method of balance charging, but the control panel has to communicate with all 3 chargers while maintaining isolation, so it is mostly used in simple low-end chargers that may be unreliable.
More sophisticated balancing chargers have an LCD screen and are fully programmable. Their balancers usually work throughout the charge cycle so the cells start to become balanced before reaching peak voltage, but most of them have relatively weak balancers. The main advantage is that the LCD screen shows you the cell voltages, so you can cut the charge rate down to help balance the pack if necessary. The display also shows how much charge is put in, so you can gauge the health of the pack.
A good balance charger may be bulkier, but will be more powerful and gives you much more control and flexibility. Many can also do Nicad/NiMH, LiFPO4 and Lead acid batteries. One charger may be all you need to charge many different devices.
Best Answer
So that you can connect/disconnect while not standing over the battery. The batteries contain sulfuric acid which is not a very pleasant substance to splash. Even the fumes are dangerous.
While connecting inside a vehicle, connect the positive first and the negative on the vehicle body while not facing the battery. The negative can be connected to any exposed metal part on the vehicle (and at a distance from the battery). Similarly while disconnecting, disconnect the negative first and then the positive cable. The reason is that the vehicle body is connected to the negative terminal. If the positive cable slips and touches any other vehicle body part, it will cause a short.