I have both the Kissbox style 801 passthrough and 510 both with the in-line battery. They're a PIA usually, LOL. Just got out my old one this morning. Battery was dead so plugged it into my computer USB and left it for a few hours. Worked great once the battery recharged.
Manual buttons (switches) on these are notoriously bad so check that if your passthrough has one. A bad battery can cause problems. If you have a cigar charger or Multi-Function charger throw the battery on that. Should get it powered up. My friend got one that wouldn't charge up on the passthrough initially, then I got her to put it in a charger and it worked fine after that.
The little 10440 doesn't last forever. Leaving it plugged in all the time shouldn't be a problem (is your computer on 24/7?). I don't know if power surges from turning it off/on would cause a problem or not. I'd suggest unplugging it before turning off the computer. Better safe than sorry.
A wall wart plug is a good idea for passthroughs really. You can get them on line (the one for $1.95 on line at usbpoweradaper.net is sold out) I use a plug for the Apple too but they're more expensive.
The inline battery it just limits the voltage to what the battery is putting out (3.7v) rather than what ever the passthrough is plugged into directly (voltage from the USB or a USB wall plug is 5v usually). The battery supplies the current that is needed for the atty to operate effectively.
Passthroughs that don't have a battery are direct wired and give 5 volts to your atty (you need to plug into a USB hub or AC/USB adapter or some other device that is capable of supplying the 5v current needed for the atty). Some have bypassed the battery box by rewiring their passthroughs or putting in a jumper wire around the box by clipping and reconnecting the wires. I haven't tried this with mine yet. They run ok for the most part (or I use my PureSmoker 5v passthrough instead).
The 10440 batteries aren't usually available locally (say from a battery shop). You might be able to get them from a few of the US battery sites but be aware that protected batteries may not fit the battery box (they don't in mine). These are unprotected:
UltraFire 10440 battery
In Texas:
Ultrafire 10440 Unprotected Rechargeable Lithium Battery - 3.6 volt Lithium AAA
UltraFire 10440 AAA Li-Ion 3.6V Rechargeable Battery 500 mAh FLAT TOP CR10440 Lithium Ion