There is another Home Depot in Queens in Bulova Shopping center out near Laguardia Airport. I'm sure the folks at any home depot wouldn't object to you quickly unscrewing the stock valve assembly of the the cooler to check diameters. Those things are built to be dissembled and cleaned.
As for gallon size, space is at a premium here in NYC, so I wouldn't recommend going big unless you will regularly use the size. To figure out what you need, take your standard batch, and standard OG (let's assume 5 gal and 1.055 for argument sake), and multiply the volume by the gravity points (5 x 55 = 275). Then, divide that number by a rough average of expected points per pound/gallon yield from grain. Let's just say 35 PPG (points per pound/gallon) are the max possible from your average base malt, and your set-up works at 75% efficiency. So your typical yield is 26 PPG (34 x 0.75). Then, 275 divided by 26 = 10.6 pounds of grain needed to make your standard batch.
Mashing 1 quart of water and 1 pound of grain will result in a final volume of 42 OZ (This is full saturation and any add'l water will only add its own volume). Using the 1qt/lb. ratio, you can expect that 10.6 lbs of grain plus your strike water, will give you a volume of 445.2 OZ (42 x 10.6). Convert that to gallons, and 445.2 / 128 oz = 3.5 gal. That is the total capacity you would need to produce a typical 5 gal batch at 1.055 OG.
This may be affected by whether your mashes are single infusion, step infusion, or decoction. So you should use your final water/grain ratio, and your standard OG to figure out what size tun you would need. For the odd time you want to go higher gravity (barely wines, etc.), you can use extract or adjuncts to add more fermentables.
For a more authoritative explanation, you may want to check out John Palmer's "Sizing the Mash Tun" in the online version of his book "How to Brew":