Guppy is one of the easiest to rare. Care to explain your tank's environment so we can help.

I'm 100% positive they don't need the oxygen pump unless you have like 1000 guppies in a small little tank.

How does your guppies die? Do you see the carcasses? Are their fins still intact?
It's quite possible that the one fish (in bigger tank) died naturally, meaning, it wasn't your fault. I doubt it's the lizard.

And yes, you must keep about 30-40% of the tank's water when you change water. And don't suddenly drain away or add in too much water because the fishes need to have the water pressure and temperature stabilized. And don't forget to add antichlorine or season/stand the water for the chlorine to get out.

And it's normal for mother guppies to give birth to 10+ fries. It varies. Depends a lot on the breed of guppies that you have.
and another advice is to change water ASAP when you see one fish with bald scales, kinda like white patches around the body. Because it's a fungal infection and it can spread. So before buying any guppies from a shop, look at the tank, see if there are any dead guppies with white spots or missing scales in the tank. Even if you were to buy a healthy one from the tank, you'll be most likely buying an infection for the rest of your healthy guppies at home.

LadyVanity,

from my experience, it better to let the guppies live in a natural environment ,  well, at least it works for me... 
currently in my tanks , i put some plants in it anacharis plants (took from the river at my kampung) ;-) try google the term for the sample pic of the plants...  and so far all my guppies seems to be fine...  and also don't use chemical medications, just an air pump for extra oxygen for the gupps....  i just put the rain water in the tank... and once the water is decreasing, i add it more with rain water ( but you must make sure the rain water temperature not too cool ) .. i think plants can neutralize all the ammonia and other hazardous thingy in the tank and also give them extra food sometimes...  currently have about 60+ guppies and bout 30+ fries...  hope this give u a little help...

i did all the 3 steps ... today i came back from work .. i found 5 guppies passed away =( ... i still dun understand why ..

HI there Syrian,

This is what also happened to me.
I bought a new aquarium for my little guppies and one by one died.
You know what is the reason?
Just like wat KimiKewl advised its all about the nitrite cycle.
When buying new aquarium you can't immediately put in the fishes.
You need to let the water run for about a  few days to one week.
After Kimikewl's advised, I googled and read all about the nitrite cycle.
Please don't buy anymore guppies like I did.
I let the water run for a few days and then only bought new guppies.
I am happy to say that I am no longer a guppy killer 
My guppies are swimming happily in my new aquarium.
They are such a beautiful sight to see.
Good luck!

i have a small tank, about the length of an A4 paper (used to keep hermit crabs, but already gone to heavens after more than a year), can i rear guppies without air pump? if yes, what is the recommended no. of fishes?

i bought with the recommendation of the fish shop owner, about 9 fishes (4M 5F)... without air pump... so far erm, so good. no signs of the fishes gulping the surface for air nor inactivity at the bottom of the tank.