Oh my. I'm so sorry you are having so much trouble. Some of them are water and cycling issues and some are very likely health issues from the various purchased fish including the original fish.
First let me try to clarify the difference between the 2 tanks. The Betta lives successfully in the little tank because you clean it out completely and change the water, likely, hopefully at least 2 x a week. They are pretty hardy fish and also have a labyrinth organ which allows them to breathe air off the surface when the water gets a bit stagnant. The little tank is too small to provide enough surface area to allow sufficient bacteria to grow to maintain a cycled tank and therefore we must change the water frequently.
The 10 gallon, as long as it has gravel, a filter and heater can cycle. It can take 6-8 weeks to go thru this process unless the tank is heavily stocked with LIVE plants. Live plants take up ammonia in their processes and your fish will not even notice a cycle.
The good bacteria grow and live on the surfaces of the tank, gravel etc not in the water itself. Fish poop, Ammonia builds up, bacteria converts it to NitrIte, which also builds up and is converted to NitrAtes. You must do water changes to dilute any numbers that get too high. (Mr. Betta would only experience Ammonia unless either of the others are actually in your tap water.)
Prime is a water conditioner that is a little more expensive but can be used to neutralize Chlorine & Cloramine, Ammonia, Ni, Na and can be used at 5 x the strength in emergencies. It should be used at all water changes.
Water testing kits, come in 2 forms. Test strips which are a little harder to read, seem cheaper up front, but don't last as long and if they get damp from humidity can read incorrectly. Many on the Forum will tell you to avoid them. The API FW testing kit with the test-tubes and bottles provide a more accurate way to monitor your tank's cycle. Make sure you shake the heck out of the bottles to get proper readings. You should also frequently test your tap water to know your parameters. Trying to achieve O Ammonia levels when testing my tap water shows .25 makes it difficult. But with a good de-chlor- Prime, and a tank full of live plants, I'm good. Major cities are known to dump chemicals into our water systems in spring and summer so it is always good to test now and then. Especially if you run the tap and smell chlorine!!
The best medicine in your cabinet is actually good clean fresh water.
I'm not familiar with the salt so can't help there. Always quarantine new fish but be aware that disease can stay hidden for months, long after you have returned them. Many diseases are caused by poor water quality. Sadly Guppies are WAY over bred and have weakened immune systems and harbor disease too.
Take your time, slow down and do things slowly to your tank. One thing at a time. Do not clean out the 10 gallon the way you would the Betta tank. You want the bacteria on all the surfaces. The only thing you will change will be the insert in the filter and eventually some gravel vacuuming. If you have live plants you should be good! Keep decaying matter to a minimum tho as that causes Ammonia. Don't over feed, remove dead fish asap, these all cause Ammonia spikes.
Keep a proper light cycle of 8-10 hours ON then complete darkness during their night. You can set it to be "daylight" very late in order to watch them till your bedtime but they need a set circadian rhythm just like we do. Remember to use fertilizers at least once a week for your plants to avoid dying leaves --> ammonia.
Hope this helps.... sorry it is so long! Keep us posted on your progress!!