1. I've tried both. Just switched back to plastic. In general I'm a fan of the 32 oz deli containers because they've got a really wide opening so it's easier to feed without spilling flour everywhere, but right now I've got my starter in a reused Talenti pint gelato container because I don't have a ton of fridge space at the moment.
2. I'm a discarder but I manage this in a few ways. I feed once a week while refrigerating when I'm not baking. When I am baking, I'll feed the starter as usual, but use what I would have discarded to make the proper amount of starter I need for my loaves of bread--this way I end up with two containers of starter. One will go back in the fridge, the other will be used for bread. The starter I keep in the fridge isn't that big--it's only 180 grams, so I'm only feeding it 40 grams of flour every week, which I think is a reasonable amount.
You could always use the discard for delicious things like sourdough pancakes, waffles, or sourdough english muffins!
3. With a liquid state, try feeding it in a ratio of 1 (starter) to 2 (water) to 2 (flour) for a bit until it thickens up again.