Welcome to ETS! Wow, that's a lot of questions.

I'm normally rather long-winded but I'm a bit pressed for time here so I don't know if I can hit on every single point you raise. The first point about any first aid for burns is to cool the burn and stop the burning process. Probably any of these products does a similar job at cooling.

Once cooling is taken care of, all of the other ingredients are, in my opinion, secondary. Aloe is basically a soothing agent and may somewhat protect the burn but I don't think it's useful on more than superficial burns. Tea tree oil has antibacterial properties, but that concerns the healing process, not really immediate first aid. Lidocaine is useful for superficial and some partial thickness burns, but for serious burns, the nerve endings are destroyed in the burned tissue so I don't think anesthetic really matters. True lidocaine allergies are very rare although I guess that's just another reason to sue a company.

I personally don't carry any of these burn gels in any of my general first aid kits, however if I worked in certain environments, like in a restaurant kitchen, welding, and a myriad of other places where burns are common, I would definitely consider it.