Technically, you can't charge and discharge a battery at the same time, as current is either entering (charging) or leaving the battery (discharging).
Can I use a battery if I'm using a power supply?
When powering it on for the first time, use a power supply if you have one. Limit the current to 3A. This will keep everything from blowing up if something was connected wrong. Once everything is working using the power supply, you can use the battery. I would highly recommend adding a switch in-between your battery and the circuit.
Can I run two power supplies at the same time?
It will just allow you to run two power supplies and have them both turn on together. For example, you use one power supply to power your motherboard and hard drives, and a second one to power your video card or cards. They'll still be seperate technically.
Is it possible to run two power supplies from one PSU?
Only if you use them to power different things, like one PSU for the motherboard and peripherals, another for the graphics card. Your idea of running 2 graphics cards from 2 PSUs is perfectly feasible. You need to keep the outputs separate so the regulators inside the PSUs don't fight one another by trying to control the voltages of the other PSU.
What if I don't charge my battery all the time?
If in fact you don't charge all the time, then why not switch your circuit (load) to the alternative power source (via a regulator) when it is available. Then you don't have to worry about the battery charging complications.
Can You charge and discharge a battery at the same time?
I thought that you could charge and discharge a battery at the same time without issue, but after googling I find that half of the articles say that you can't do that (or you can, but the battery life is shortened or the battery will burn). I also came across the term pass-through charging. So what is the answer to this?
Can I use battery power while the charger is charging?
When the charger is charging, the charger is providing the power to the radio and some more power to the battery. You can not possibly use battery power while the charger is operating. The current will not go into the battery and then come back out of the battery to run the radio.