Want to cover several things so going to break it up into sections. Sorry for how long the post is, but you appear to be new to this and information is always good. I'd strongly recommend doing a lot of research before getting anything else. Read the stickies in teh sub sections of this fourmto get you started, especially::
Most of those are for tank cycling but its good to read them all.
What were you doing for acclimation? If its just float the bag/cup for ___ minutes then dump into the tank thats not proper acclimation and your fish likely went into shock from the drastic change in water parameters. All fish need to be properly acclimated slowly and gently. Getting there water to tank temp range via floating is only 1 part of it. The other is slowly adding small amounts of the tank after to their container (you can either do this while they float or add water while set next to the tank then float once that part is done). Why? Because fish are sensitive to changes in pH, temp,and nitrogen levels. If a betta goes from a store cup (which I can guarantee you is going to have off the charts-over 8ppm ammonia) to a newly set up tank ( usually 0ppm ammonia) they can go into shock and die within 24-72 hours. You also don't know if the store's water is the same pH as your own either. SO gradually adding water is important to get them use to the new water safely.
There are 2 methods of adding water for acclimation. The first is more common (all you need is a clean small cup/bowl/container-rinse it out thoroughly so there's no soap residue left first) cupping and adding very small amounts of tank water ever 5 minutes or so to the betta's container. DO this for 30 minutes at least (more is better) you want to add tank water until the ratio of tank to container water is 3:1) -if the container the betta is in is only 1/4 full you want to gradually add water until it is full. If their container is getting full carefully remove some water from the betta's container and continue adding tank water bit by bit.
The other method is using an airline tube for "drip acclimation" if you have airline tubing for an aquarium air pump around its an easy method. You ca put a knot in the tubing so water drips out slowly, or use a little plastic valve to adjust flow (used to adjust strength of air let through tubing-tetra whisper air pump comes with this valve but you have to buy the airline tubing separately). You put none end into teh tank and suck on teh other to start a siphon. The bettas container will have to be lower than the water line of the tank to keep the siphon going. Adjust your tubing so water drips out at abut 1-4 drips per a second and place the other end not int the tank into the bettas container. Keep a close eye on it to prevent overflow (remove water if needed) and let acclimate at least 30 minutes (againts same ratio of tank container water ad mentioned above should be reached before finishing this step). Once done drip acclaimting remove the airline tube from the main tank and betta container and float the betta's container for 15-30 minutes (longer is better) to get it to tank temperature.
No matter which method you use to acclimate I'd strongly suggest getting s fish net and netting the fish from its container to put into the tank (quickly) when acclimation is finished. You don't want to add the nasty store water to the tank by dumping it in with the fish. Dumping the store water in will raise the tanks ammonia levels and you don't want that!
Firs toff your "moss ball' is marimo-its not a true moss but a specialized form of extremely slow growing hair algae. As an algae you much be very careful about any chemical treatments, anything that kills algae (bleach, hydrogen peroxide, or potassium permanganatedips, excel, or any algaecide specific products) will kill the marimo. Unfortunately the first 3 dips are the most common ways of disinfecting plants from disease, so I'd recommend just quarantining the marimo in a jar with some dechlorinated cold tap for over a month. Marimo is very resilient and can be kept in a very cold water and have very little light (it actually is better off in the shade with no direct light), one site's recommendation for marimo care says you can keep it in water in the fridge during hot summer months if your tank is going to get over 80f range. Btw slow growing=slower to absorb nitrogen (ammonia/nitrite/nitrate) marimo grows so slowly that a tank full of marimo (I'm talking barely any room for a betta in the tank) won't be enough to make any noticeable dent in the nitrogen produced by the betta
The amazon sword can be dipped but if you are only worried about ammonia hurting the sword there is no need. Plants do consume ammonia (though they eat up nitrate quicker) but 1 plants won't magically absorb all the ammonia of a dead snail (they produce a lot of ammonia especially for such a tiny tank-larger tank=more water volume=more diluted ammonia). Do you remember learning about photosynthesis at school? Same thing for under water plants. Their main limiting factor for growth is Co2 as water does not contain as much as air. But plants also need light (they can't photosynthesize without it), and other marcro and micro nutrients (ferts). The trick with planted tanks is balancing all these (lighting, co2, ferts, and nitrogen), having an unbalance will cause deficiencies in the plants leaves, algaes, or possibly death of the plant (example: too light light to properly photosynthasize the plant can starve over a few weeks time... too much/strong a light and you get hair algae...). I can go off on a huuuge tangent about plants but first let me say: an amazon sword plant gets way too huge for a anything smaller than a 20g tall. they can get nearly 24"x24" tall and wide. If you want to keep it in a small tank you'll need to trim its leaves as they get to big (cut off near base of plant) also to keep it going: a good grow light and a fertilize (easiest thing is a root tab made for aquariums, also get an iron specific fert as sword are iron hogs). Water movement (from a filter) will pull co2 from the air into the tank-but don't go crazy on a strong filter-bettas prefer low/little flow.
If the snail if dead you'd know it as soon as you removed it from the tank: they smell SO BAD you don't need to sniff it closely, it just fills the air with foul. Nerites have huge bioloads (aka poop alot) and are probably the most picky eaters of the snail family as they will not eat algae wafers and typically won't eat blanched veggies. Because of this they need a plentiful suppl of algae within the tank to keep them from starving. A brand new tank will have no food for them and they will starve. You can make algae rocks by taking inert and ideal smooth rocks (river rocks work well) and putting them in a bucket/whatever with dechlorainted water and setting them in the sun or under a grow lamp to grow algae-you'd need several rocks as the snails can clean them faster than the algae regrows in the rock container. Also they should never be added to a tank tank that have not be cycled. And Finally I'd personally not recommend a nerite snail for anything smaller than a 10g tank to deal with their bioload and to give enough surface area for algae to grow on to give them a chance.
If its not dead take it back and ask for a refund. Shrimp cannot survive uncycled tanks, they're extremely sensitive. They do best when added to a well planted and already cycled tank that has been up and running over 3 months to allow their natural food to grow in the tank. Also there is the possibility of them ending up on the menu if housed with a healthy betta (one of mine killed ghost shrimp an amano shrimp that were as big as him, guarded the corps and ate it for a bit before I removed it).