I am also very glad you posted. You did some things wrong, but you did one thing right enough. We try to tell people it's dangerous enough to take a few precautions, but a careful person will not likely have a fire.
I'm not clear whether you had the batteries on the bike or not, but you clearly had them on a fire resistant enough place to avoid a major catastrophe. VERY GOOD!
No matter what you are doing to charge and balance, or how much you spent on the balance boards or chargers, charging where a fire is less likely to burn the house, car, kids, or neighbors is the single most important thing you can do. Personally I charge off the bike, but the main thing is don't charge next to something extra flamable. Those 2x4's got charred, but at least it wasn't the paper recyle bin or the gas can right next to the charging packs. Give that charger a few feet of space before anything that can burn is present. Smoke alarms near the charger, and I have a flat shovel nearby so I can flip the pack outside.
Your ready to hand non flamable blanket worked great too, I need something like that too! I even have some Nomex fabric in my garage, why didn't I think of it?
Very simple precautions can help a LOT. An old sink is not a bad idea, but just something less flamable than wood or carpet can work fine. Sheet tin, sheetrock, tile backer board, brick or tile can serve to make a less flamable wall and floor next to your charger.
What you did wrong has been pointed out, but I'll repeat for the noobs and lurkers reading. The battery medics ability to balance the pack was slower than your charger. Idealy, a "bms" will discharge the highest cells fast enough, but in addition to that, the charger or power supply shuts itself off before any cell gets too overcharged. Then when the voltage of the entire pack drops enough, the charger turns back on for just a brief period to keep charging the low cells. This on off on off cycle repeats many times till the pack is balanced.
It takes a long time, especially if your pack is really out of balance. So that's why you were waiting so long for it you stopped monitoring what was happening to the high cells. You can't sit there watching battery medics your whole night. I've done the same thing myself to a pack I was experimenting with, after watching for a hour went to pee, came back and a pack was overcharged and ruined. Wow, that was quick!
Now I advise people with RC lipo to get at least one RC charger. It's just fast, safe, and convenient to charge a single cell to balance a pack if you have a very large inbalance. Other balancers work fine when a pack is just slightly unbalanced. But when really badly unbalanced, I think it's just better to simply charge only the low cell. That way the rest of the cells don't have to risk an overcharge at all. What I mean is this, a bms can't make a mistake if you are single cell charging. Only the cell being charged is at risk of overcharge.
Single cell charging to balance the pack is the only way to go when you have one cell at 3.1v, and another at 4v. That's just too much for most balancers to handle in any reasonable lenth of time.
The safe way to single cell charge, is with a good charger you trust to shut off at the right voltage, such as an RC charger set to 1s charge. You just set the charger to 2 amps or less, and use the jst balance wires to hook up only the lowest cell. At 1-2 amps, the jst plug and wires can handle the load fine.
The other option would of course be to single cell discharge the 4v cells, with something faster than a battery medic. A turn signal light bulb, or again use the RC charger which can also discharge at 1s.
Lastly, NEVER hesitate to chuk into the recycle pile any pack with a cell that won't perform. That cell is the risk that you don't need. For a savings of $40-60 you risk your house? GET RID of packs with cells that continually get way out of balance.