As you rightly guessed they die off because they aren't being looked after correctly. You can't just chuck plants in and leave them to it, they will eventually die off, they need feeding as well. There are many routes to go down with planting but if you just want a few plants for some hiding places and to help keep nitrates etc down(not an alternative for water changes / proper maintenance) then I'd suggest getting some fast growers & some bottled plant food. If you decide you want to get serious with planting and want to go down the Co2, E.I route etc then read up about it all on the
plants section & get some ideas of what you'd like your tank to look like / what you can afford to do in the
journals section.
As for your tank size it's not big enough for all the fish you have. Do you have plans to upgrade very soon? Your correct your tank is roughly 27.5G but that's not taking the decor, gravel etc into consideration it's probably more around the 25G mark but either way it's too small for some of the fish you have.
The bichir needs a 30G min, but at a push your tank may be ok, I'm not sure about the actual dimensions a bichir requires(I only owned them for a short time in a 66G so space wasn't an issue)
The common plec gets waaaay too big for the tank you have, ideally it needs a 4ft x 2ft tank(but some people keep them in slightly smaller tanks)
The fire eel again gets too big for the tank it's in, ideally it requires a 6ft x 2ft x 2ft as this has the potential to grow to 24inches(which isn't common they usually don't get to more then around 20 inches)
The tiger barbs & corys are fine in the tank you have(depending on exactly how many you have?), the corys, bichir, eel & the plec would benefit more having a sand substrate rather then gravel. If you can't upgrade your tank very soon then I suggest you rehome the bichir, eel & plec. If you do upgrade you need to keep the corys in a different tank to the bichir as it will at some point try to eat them, I'm not sure if the eel would have a go as well, I've never kept them and don't know very much about them. Is the picture above the tank they are all in? Just I can't see any sign of any of the fish apart from the corys and what looks like a gourami in the top left corner and some sort of small fish in the top right.