I know this is an old question, but I've been doing some similar things lately and so I'd like to give you a different perspective. First of all, I fully understand why you're selecting these boards and cells on e-bay: to save money. I'm with you there 100%, and I think you obviously know the risks. There are good buys and bad ones as well as mis-representations, and as long as you understand this, the low prices do justify the experimentation. That said, let me answer directly...
I have some experience with battery management chips and circuits, and what you're suggesting "should" be fine, though you would need a supply that at least can be adjusted to a little higher, like 12.6V. Understand, however, that some inexpensive switchers (like wall adapters) can generate a lot of ripple, which I have found can confuse BMS circuits. If the BMS can't do exactly what it was designed to do, all bets are off, and you may never know. So if you use a switcher, consider adding some extra filtering. Alternately consider a higher voltage with an analog regulator, as it will provide current limiting. Its hard to know what inward current limiting exists on some of those e-bay BMS boards, and Lord knows you won't easily carry on a technical conversation with most of the sellers. But current limiting will give your batteries a bit more longevity, at the cost of a longer charge time, but in any case the current must be known by actual measurement. And be aware that picking a 12V 2A switching supply does not guarantee the battery and BMS board combination won't attempt to draw more, which could damage the supply too.
To remedy that, you can often get away with a higher voltage, even without a regulator, using a simple resistor ballast, and solve a multitude of problems. As long as there is something to limit the current. The reason is that these BMS boards effectively go to open-circuit, thus isolating the battery, once full charge is reached. I've verified this many times with a bench supply set to 16V with a 10 ohm power resistor in series, with using the exact same BMS board you linked! The voltage at the input/output point clamps down to under 12V until the battery reaches full charge, and then springs back to 16V, while the cell combo sits at 12.6. Granted, this is not the fastest charge, so eventually you can go to a lower resistance. Sometimes an auto turn signal lamp works well in this kind of circuit, because the bulb can offer an indication of full charge when it completely goes out. I know that sounds like a kludge, but what I'm saying is the BMS board will give you a lot of latitude for making a charging supply, as long as you're careful and measure whats going on while experimenting.
Finally, while we want to avoid repeating other people's answers here, I will second that you be doubly mindful of safety. When experimenting with inexpensive "e-bay" or "ali-express" Li-ION cells, BMS boards, and home brewed charging systems, always be aware that things can go horribly wrong, causing burns, fires, and more serious injury. Do all initial charging in safe, fireproof environments, with adequate protection to yourself and surroundings.
That BMS looks identical to one that I've used and it does NOT do balancing or proper charging. Just protecting from over current, over voltage and under voltage - all of which are very important.
You will need to add balance wires and get a balance charger for this battery.
Balance wires are easy - they go exactly where the bms wires go - one to each pos and neg of each cell. The balance charger will have a slot for those wires.
I recommend looking at the IMAX b6 charger because it is the cheapest balance charger I have seen and seems to work reasonably well up to batteries with 6s or less.
Best Answer
If it is current-limited, yes. Otherwise no.
If it is truly a charger, yes. But be aware that some people mistakenly call an AC adapter "a charger". An AC adapter is not a charger.
Ideally yes. In practice, any current-limited supply with a top voltage of 12.6 V (or adjustable to 12.6 V) will work. Just make sure to disconnect it when the battery is full.
A Li-ion charger may include a shut-off function when the battery is full, something that a power supply doesn't have.
Yes, but for a single cell. It can't be used for a 3S battery.
Correct.
Yes, you do need both a charger and a BMS. But, no, the TP4056 is not a charger for a 3S battery. You need a 12.6 V charger.