[jdluedtke]

We have a brand new 20 gallon water heater. We changed the element to a 240v (electric was originally set up for a 30 gallon tank which was 240v so we decided to change the element to heat the water up faster). However, we noticed that the cold water pipe is connected to the hwh where on the hwh it says "hot" and vice-versa. Our hot water runs out very quickly (yes we understand this is a small tank) but if the pipes were installed backwards, could this also be causing a "hot water problem"?

[speedball1]

if the pipes were installed backwards, could this also be causing a "hot water problem"?

YES! Ya got to understand how waters work. The cold side has a dip tube,(seeimage0 that directs the cold water down to the bottom of the tank where but's heated and raises to the top.
If you don't connect to the correct side you're allowing cold water to go directly into the hot water waiting to be used. I have a video I want you to watch. Click on. How a water heater dip tube works and what it looks like.
To address the size of your heater I have a 30 gallon one element water heater that furnishes more then enough hot water for two people. NOTE and WARNING! DO NOT apply heat directly on the cold water connection. The dip tube has a plastic flange that secures it to to cold water connection. If you apply heat directly to the cold water connection you'll melt the flange, the dip tube will fall in the tank and you will get a shortage of hot water just like now. Your solution would be to come out of the heater with 3/4" brass nipples band connect the water to them. Good luck. Tom