Emergency, Concerned For Life Of 5 Tropical Fish

So my biology teacher announced to the class that he was giving away fish, or else he would flush them down the toilet. Fearing for their life, I offered to take them. However, I have little prior experience in fish keeping. I didn't have time to cycle the tank at all, and all five fish had to be put in at once because they were getting ill. Two of them have ick and I have treated the water with the medication. The water has not had time to heat yet so it is very cold. The filter is air stone run only but we are getting an air pump tomorrow. I am seriously fearing for the life of these fish, but I have no idea what to do :c Advice?
(Also, I do not know the breeds of the fish but I suspect one if pregnant. Their original environment was a dirty, sparsely filtered wasteland of a 5 gallon tank. My tank is ten gallons with gravel. The fish were gasping for air but seem to no longer be doing so.)

Post pics and we can help you a bit better.
 
 
Things you'll need:
 
Test kits - ammonia and nitrite being the most important right now.
Proper filter - hang on the back would work nicely - fairly cheap.
heater - especially important with ich.
water dechlorinator - you are in for a LOT of water changes.
 
 
The top priority is actually the ammonia the fish will be producing, not the ich.  You need to treat the ich while doing the water changes which will be a little trickier than just water changes.  You will need to replace the medication you are removing.  So, if you have 5mL to the tank to treat the ich, but then have to do a 50% water change - immediately afterwards, you need to add half the original dose (2.5mL) to the tank to account for what was removed.
 
 
Are you sure its ich?
 
 
(Can I add that I really abhor stories like this?!  A biology teacher should have more respect for the animals than that, but it is far too frequently seen here and other places, I'm afraid!)

Right now your main concerns should be
Temperature
Filter
Water quality
Proper feeding
Water dechlorinator.

Once you have the filter and heater set up I'd do water changes daily, how much depends on the type of fish...what type? Can you give us anything to work with? I mean five fish that used to live in a 5g is really all I have to go off of, what size are they, schooling fish, cichlids, catfish?

   Tank where fish are housed
actual fish (sorry, Calliope is camera shy or something, she keeps bustling her way around everywhere anytime i try and take a picture)
 
It's almost certainly ich, one of the fish has a large white growth above its eye and the other has it on its tail. The filter should be upgraded to have an air pump by tomorrow. As for types, all I know is they are tropical and one (the blurry one) is blue spotted while the others are more of a reddish bronw.

That looks like an Undergravel Filter.  They work just fine.  BUT, the plate needs to be ALL the way on the bottom of the tank - straight on the glass.  Then, there needs to be a nice layer of gravel over the top of it.  The gravel itself will become the filter media.  
 
 
It does need a fairly strong air pump.  Good to know that you are getting that squared away.
 
 
Next, I can't make out the type of fish that you are deaing with, but they all appear to be the same.  How big are they?
 
Finally, the "ich".  Would you describe the white spots as looking like grains of salt (or sand) on the fish, or is it less defined and kind of fuzzy?
 
 
 
While you are picking up the air pump, pick up an ammonia and nitrite test kit as well.  This is essential for the process you are embarking on.

Hmm, could you get a picture from the side if possible? Someone more experienced might be able to identify those from that picture but I can't quite see...

I would say a mix of both. The main spot on the small fish is sort of fuzzy but there are white grains all over the larger fish. 
Four of them are the same fish, one is a larger blue breed.
I will fix the filter asap if  that will help them. 
 

It will make for a better flow of water/nutrients over the gravel.  The more water that flows over them, the better.  

Okay, under gravel filter has been fixed. I'll see what I can do about getting a side pic of the fish, they're becoming more active as of fixing the filter.

Does the "bigger" blue one look like this:
 

 
Or similar?
 
 
More likely this one:

Sort of, here is a better picture:
and here are what the small ones look like (although this one is significantly fatter, not sure why)

Opaline or blue gourami.

Gourami! That's the larger one, the gourami. I don't know which one the fat one is, though.

The top one that is, looks like an opaline

I suggest investing in a cheap hang on back filter as they're much easier to maintain than a under gravel...I'd suggest going to your fish or pet store and finding one that's rated for 20 gallons or 30.

The other appears to be a rainbowfish. They are active swimmers and are not at all suitable for a 10 gallon.

If the gourami is a dwarf, like the ones I showed above it will be ok in a 10. If its an opaline, it too needs more space. I commend your efforts, but you'd do best by the fish if you reformed them to a proper sized tank, as soon as you can.


Opaline:


Is this the other fish?