Rotten egg smell in water is usually the smell of sulfur. It's very toxic to fish. You can buy filters for sulfuric water but they can be pricey. At this point the distilled water is going to be a million times better than refilling the tank with sulfur water. If you can, try buying a rain barrel and doing water changes with rainwater.
Whitish colored ghost shrimp are either sick (infection) or about to shed their skins. Dead shrimp are pink. Sometimes you may see bits and pieces of white or clear shrimp skins lying around after they shed. They usually eat the skins right after a shed, but sometimes they can be mistaken for dead shrimp.
Shrimp jumping out of a tank usually only occurs when they get startled or chased by a fish. If you have a fish in the tank that likes to chase the shrimp, they may be jumping to escape and end up on the glass. It happens very often here at the fish store when we are feeding the fish in the shallower display tanks.
Another possible reason they are jumping out is poor water quality. I would invest in a liquid test kit. Don't get the strips as they are unreliable. You can take a sample of your water to a pet store to get tested, but they'll use the strips which won't give you an accurate result.
Because of the sulfur your water may be super low ph. Ghost shrimp prefer a neutral ph usually.
You may also have a copper content in your water which is toxic to invertebrates, but one of the above problems is more likely.
The only real way to find out what is going on is to buy a water test kit and have your water tested. Sometimes you can take a water sample to your local water authority and have it tested for sulfur, copper, magnesium, and related things for free. You also want to test your water for ammonia, nitrite, and nitrate.
I don't suggest changing filter pads entirely all at once. You should rinse out the filter pad once a week or so with tank water, and then put it back. You can replace the filter every 6 months to one year, but you should add the new filter pad in and leave it in for at least a week and a half before removing the old pad, to give the cycle bacterias time to pollinate the new pad. Ideally, you should have a packet of biomedia, activated charcoal, and a filter pad all at once. The biomedia will host the majority of your bacteria (ceramic balls are often used), the activated charcoal will absorb heavy metals, excess acids, and tannins from the water, and the pad will trap the larger bits of debris.