Well, it certainly doesn't appear to be overstocked with so few fish in 80 gallons!
First, you don't have enough water turnover using such a small filter, although with so few fish, you could just about go filterless (not recommending that though). If that hang on back (HOB) filter is all you have, then every time you change the cartridge, you basically reset that filter to scratch. You don't need to go get anything expensive, but you'll want a filter with more capacity than that little filter.
I'll assume the tank doesn't have live plants? My guess is when you feed, you overfeed. That's not intended as any negative, as I know I still tend to overfeed my fish - most of us on the forum probably overfeed!
Here is what I'd suggest in the absence of any water parameter measurements:
* Go ahead and do a good size water change (50%+ new) and fill the tank up. I'm not sure how you can run the HOB filter if your tank isn't filled up all the way, as you indicated in an earlier post. I agree with what Byron said about the bacterial bloom, but at this point, lets just get reset to a known point.
* as you do the water change, either use a gravel vacuum device of just get your hands down in the gravel/sand and stir it around vigorously. If you get a big nasty cloud of "stuff" when you stir the gravel, after the tank only being set up for 3 months, then you are overfeeding. Siphon out a good bit of this stirred up organic matter. The other option is to stir it up, let it settle and then suck it off the bottom as you do the water change.
* having a dead fish in the tank a couple of days will cause a problem if you don't have enough filtration. You don't have adequate filtration for that tank size. Plan on adding some more filtration. It could be a second, larger HOB, a large sponge filter that's driven by and air pump or powerhead, or if you have the $$ a canister filter is an option. For teh fish load you have, you actually could probably get away with the single filter you have if you are careful not to overfeed.
* Regarding the fish deaths. Unfortunately, when purchasing any new fish, you have to assume it is infected with all sorts of diseases and parasites. This is particularly true if you buy from stores that "move" a lot of fish like the Wal-Marts and big-box pet stores. It's not a negative on them, they simply run so many fish through their tanks, that disease transmission is inevitable at some point.
You need to either A) set up a small (10 gallon) quarantine tank and/or B) treat your big tank for internal parasites and external parasites. NOTE: this advice goes counter to those who advise against the "shotgun" treatment. It's not what you normally want to do, but I'd bet you have something nasty in the tank. I had a huge problem with fish deaths (lost over 13 juvenile Angelfish until I started quarantining and treating for parasites as a routine precaution).
I'm sure others here will have more to add, but that's my $0.02 worth! Good luck and be patient!