I think I made a big oops...

Let me start by saying that I am very new to brewing. Two days ago I set out to brew my 3rd and 4th batch ever. The third batch was a gumballhead all extract clone recipe which is doing fine. My 4th batch, the centennial blonde all extract version may be in trouble. I made the rookie mistake of pouring hot wort in a BB carboy. It kind of morphed the shape as I did it. I filled it with the cold water portion of the recipe capped it and set it aside to ferment. I'm not sure if it actually melted or if it just warped. I thought that the pressure from fermentation might expand it back to normal shape and size. I left both batches alone and left my house for about 48 hours. Upon coming back home I found my gumballhead is bubbling away. The blonde had blown it's top I cleaned everything up and put the top back on. It has been a half hour and it is not bubbling. It smells great BTW... The carboy is still in it's warped shape. Did I ruin my carboy? More importantly did I ruin my beer?!?!? Either way I'm glad I have a batch of gumballhead doing well and a keg full of brown ale currently carbonating!

I've had an airlock and stopper blow off before and I just cleaned it, sanitized it, and put it back on and the beer turned out fine. With that much gas escaping from your fermenting beer not much is going to have the chance to get in there and set up shop. Chances are good the beer is fine.

I've never used a better bottle before, but I'm guessing that should probably be the last beer in it. I don't think I'd trust the integrity of the plastic, plus fermenter shape can actually have noticeable effects upon fermentation (which might be one reason you had that crazy blow off). Though maybe someone much more familiar with Better Bottles can chime in here.

If the pressure does not force the BB back into shape I would say it's done for. What temp was the wort when you pitched the yeast? And are you controlling the temp now? These steps would tell if the beer is going to be what was intended. Unless you get an infection you will have beer.
You have fermentation or the stopper would not have blown out. If you started hot, the fermentation may have finished already. As has often been stated the presence of a bubbling airlock is not a reliable indication of fermentation. You will need to take gravity readings to determine that.
Also I guess the stopper seals well since it had enough pressure to blow it off but if it does not seal well it may be a problem after fermentation slows and co2 production lessens.

I would guess that if your wort was hot enough to DEFORM the BB, which is fairly sturdy, even with adding your cool water you may have pitched way too hot.

I also don't think that your blown off bubbler was NECESSARILY the result of fermentation. Unless you measured your pitching temp, the blown off bubbler could be the result of the very hot wort heating the air in the carboy and expanding it.

I'd almost be afraid that your yeasties were dead, rather than too active. If you're sure they're fermenting, RDWHAHB. Make sure the rest of your fermentation takes place at a proper temperature, though.

As for the bottle, it doesn't need to take any pressure, so if it stands up and holds wort, it's totally usable. This batch may or may not have a plasticky taste though, like when you leave a water bottle in your car in the sun.

+1 for hot pitching. Add more yeast and it might restart but I'd be worried about err, plastic poisoning...

I would say that fermentation shape matters quite a lot when dealing with preasure at large volumes, your problem is that if it was warped inward, it's possible you just don't have any head room left for the gas escape from and you had a blow out due to a large buildup of krausen and gas. I would call it quits on your better bottle, the heat isn't going to be the problem as they are made from food grade plastic, but I think the head room might be a huge problem. Be very happy that you did not do that with glass, you would have broken the bottle and had glass shards covering your floor.

What comes out, I am guessing, will be beer and you did not ruin your beer, but I would get another better bottle or a 6.5 or 7g plastic bucket.

I didn't pitch the yeast until the wort had cooled to 75. I pitched the yeast directly into the carboy after it had cooled. I know it fermented from the amount of krausen I cleaned up... lol I'm thinking that the beer might be ok but as others have said the bottle may very well be done. I'll try to expand it back out when it's empty but aside from that I'll be ordering a new carboy and a few more ale pales... lol

If it deformed the plastic it is finished as a fermenter
You are probably ok drinking the current batch but i would not recomend continued use