Hi Rendy,
It's no trouble at all, I'm happy to help. You are right, even though deep cycle batteries are more resilient than typical LA batteries, they should be recharged before they reach zero. It can get tricky trying to determine the remaining charge of a battery. There are ways to
estimate the charge based on voltage, but these are estimates only. You can buy advanced battery monitoring tools, but I feel that the voltage estimates are fair enough. The inverter I got, which Glenn linked to a few posts above, actually has a volt meter built in, so using the chart in that link I posted you can actually get a decent idea of how much power you have left. Now, bear in mind, each battery is different and as they age, capacity at certain voltages might be different, but it's still good for just getting a basic idea of the charge remaining.
As for using the battery while charging, I don't have much experience with this. Typically, I would think that if you had a way to charge the battery, you would not need to use it, but I think it would be possible to charge while using. You would likely have to get a special kind of charger with some advanced circuitry though, as I'm not sure the Battery Tender would be designed for this. You would also have to make sure the charger you got would be able to charge the battery as fast as you are discharging it.
Glenn, I'm still very happy with the inverter I got. Thankfully I only had to use it once so far.. haha. I chose that one because if I did get more batteries or set up a grid, I would be able to power much more than just my basic pumps with it, enough to give the tank a few hours of light if I really felt like it. As for it only being modified sine wave vs. pure sine wave, I haven't had any problems with that so far. Really, from what I've read, MSW inverters are fine for everything but advanced medical equipment.