Would placing the fermenter tank in a tub of water be a good way to handle hot environments?

As mentioned before, using a swamp cooler is an excellent idea. In addition, if you work long days like me, and have no AC, like me, and your ice has dificulty maintaining the cooler themps you want, like me... you can add a small fan or even a box fan, blowing on a beach towel draped over the fermentation vessel thats dipped in the water. this adds additional evaporation causing the towel to become a cool layer of insulation.

This method definitely works. I have used it many times in Australian summers, with the water temperature hovering around 23degC, regardless of air temperature swings. I have never resorted to adding ice.

This method is sometimes referred to as a "swamp cooler", and is well known and used in homebrewing circles. Honestly, if the brew shop employee told you it wouldn't work then they are either (a) trying to sell you a brewing fridge, or (b) not that educated on homebrewing.

Change out some ice packs in the water twice a day and you get get down to the low 60's for the fermenting temp. For a 5gal carboy, I think you'll want the water in the garbage bin to reach about the 4gal line. Be sure to stick a thermometer on the side of your carboy/bucket, at about the 4.5gal mark, so its above the water level, and below the wort level.

If you are worried about mold and mildew, you can add a tbsp of bleach to the water every day or two. That should keep the water fresh for a while. Plain water isn't really going to attract the really nasty bugs like acetobacter, so your vodka airlock will be more than sufficient to protect the beer.

Temp control is crucial to beer quality for about the first 3-4 days, and less important after that (for ales anyway). Make sure you are super vigilant about changing the ice packs (frozen 2-liter bottles work good too) for that time, and then after that, you can relax a little and maybe just change them out once a day or two. I wouldn't let the temps get above 72-74F, if I could help it.

If you don't want to swap out ice packs, drape a kitchen towel or a t-shirt over the top of the carboy so that about half of the fabric is in the water. Put a small fan to blow directly on the shirt. The shirt will wick water up and the evaporation that occurs will steadily hold down the temps too.

1) Can I just place my fermentation tank in this tub of water to counter the heat? Yes. This will work to a degree (ha, ha.) The water is slowly but constantly evaporating. The energy need to make liquid water into gas comes out of the water's temperature. This "evaporative cooling" will help cool your wort by a few degrees.

2) Will this method work during the summer, when I have a 90-95F day? Not really. You'll see the same absolute drop in temperature. If the evaporative cooling takes an 80 F. wort down to 75 F., it will take a 90 F. wort down to 85 F. I've solved this problem by freezing water in 1 qt plastic bottles and adding them the to the water bath twice a day.

3) Will bacteria that collects in the bucket of water be a greater threat? Not if you're careful. When you're ready to remove the beer from the fermenter, take the fermentor out of the bucket. If you ferment in glass, clean the neck of the carboy with no-rinse sanitizer (vodka will do, but StarSan is cheaper). Don't try to rack the beer from the fermenter while it's in the water bath. At some point, you'll have removed enough weight for the fermenter to become buoyant, and it will tip over, perhaps letting water in, and definitely disturbing the sediment.

Absolutely. If you need to cool the bucket further you could alternate adding ice packs to maintain your fermenting temps.