First, welcome to Tropical fish Keeping forum. Glad you found us.
You have asked quite a lot, so pardon me if I skimp over some things and concentrate more on what I see as likely to be issues. And the first is your mix of fish. We have profiles of fish and plants, it is the second tab from the left in the blue bar across the top of every screen. You can find information on many (I would go so far as to say all of the commonly-kept) fish including tank sizes, numbers of fish, compatibility, water parameters, etc.
Tetra and barbs are shoaling fish, which means they must be kept in a group. Six is the usual minimum number, but some species need to be in larger groups due to aggression, and I'm thinking your "barb" may be in that group. But before jumping into that, a 10g tank only has so much room, and this is not only physical space for the fish to be themselves, but water quality is affected; the more fish in less water, the harder it is to maintain a healthy aquarium and the fish will always be stressed and therefore liable to disease issues--just like humans. Before buying any more fish, you need to research what you have, decide if you want this or that, research other suitable fish, and decide how to proceed.
The losses were undoubtedly due to the nitrification cycle, which is explained in a sticky at the head of this section of the forum; http://www.tropicalfishkeeping.com/f...m-cycle-38617/
The first thing to know when selecting fish is your water parameters; it is safer to select fish that will manage better in your water (hardness, acidity/alkalinity) than it is to attempt adjusting hardness or pH. I won't even start down that road, it is too complicated at this point in the discussion.
Briefly to plants; live plants are a great benefit. Water quality is somewhat more stable and easier to maintain as such with live plants because they filter the water incredibly. They are not difficult, but as with fish, you have to understand their needs and provide the basics. There is a series of "sticky" articles at the head of the Aquarium Plants section entitled "A Basic Approach to the Natural Planted Aquarium" that will provide you with a lot of basic info; don't let it scare you. Once you grasp the entire concept, the individual parts are not difficult to provide. Lily bulbs are not the easiest plant; experienced planted tank aquarists often have trouble with keeping them alive. We can discuss more suitable plants later, once you've worked out your fish. On that, I would suggest you consider small fish in a 10g; with plants, you can have 15-20 of the smaller species thriving, providing lots of colour and activity; as opposed to 2 or 3 fish. There are some small species under Characins, esp the Ember Tetra; and under Cyprinids, there are dwarf rasbora (Boraras brigittae), Dario dario (Scarlet badis), the Celestial Pearl Danio--not to mention dwarf catfish like pygmy cory for the substrate.