Do I need a sump pump in the new construction sump pump pit

As kkeilman points out, your pump is fully submersible, so that part should be no problem.

You can easily just piggy-back a float switch like this one on your existing pump. It has a special plug that you plug the existing sump pump into (no wiring necessary), and this one in particular handles up to 1/2 HP, which is more than enough for your 0.3 HP Zoeller.

The nice thing about these tether floats is you can adjust the heights depending on where you attach it, and how much tether you give it. Just be sure to test it manually moving up and down to be sure it's at the right heights, and it's not going to get hung up on anything (causing it to either stay on, or not turn on when needed). I highly recommend you secure all the wires going into the pit with zip ties or electrical tape to be sure they're not going to fall down in the future.

You can leave the existing float working, which is fine - it will act as an extra safety to shut it off if the pit is dry, but also be sure this one doesn't get hung up - because both floats will have to be "on" to make the pump turn on -- if either is off, the pump is off.

If you don't want to use the existing float, there may be a way to bypass it internally by rewiring the pump, or you can simply zip tie it "up" so it is in the on position all the time.

The water should not be that high in the sump. It should be down to around the 2-3" level inside. This is what I'd call a safety margin which allows the pump time to rest should there be any sudden surge of water coming in from the drain tile or where ever else it might come in from (only other place I can think of off the top of my head would be the A/C drain, but you wouldn't get a surge from there). If a pump has to pump continually without resting, it will burn up the motor faster due to heat build up.

There are two things to test. First, find the float switch which should turn your pump on when the water gets deep enough (it should be a ball floating in the water somewhere, maybe under the drain pipes). Rattle it around and see if it will turn the sump on. If it was stuck under the pipe, it may just need to be dislodged. If you rattle it around and the pump does not kick in, it could just be bad float switch, which you can get a replacement for if the main power cord runs through the check ball cord (hard to explain, but there would be a plug within a plug as it plugs into the wall).

The second thing to check is if the pump itself is good (if the check ball does not turn it on). If there is the plug-in-a-plug situation as I described above, pull the two plugs out of the wall socket, pull them apart, then put the pump one (should look like a regular plug) back into the wall and see if the pump turns on. If it does, you can get a replacement float switch and you should be golden for the mean time. If it doesn't, you'll need to replace the pump.

Unless there is a longer manufacturer's warranty on a pump, you should replace them every two years anyway. I would bet this one has not been changed for quite some time. It gives a little piece of mind in doing so. As well, if you do not keep the maintenance up on your pump and the a flood ensues where the pump could have handled the difference, the insurance company might not pay for cleanup and repairs, so this maintenance item is a must in my house.