Charger cord lost the cord that charges the battery

Hi, I don't think the change in the charging cable caused the problem, the fact that the battery is old makes it prone to such issues.
It's always advised you use the cord that is supplied by the manufacturer or make a like-for-like change this ensures the device will perform optimally, having said that, I am not sure that a change in the charging cable would produce such an issue instantly.
That's why I insist that you may need to replace the battery or check if the battery works or behaves in the same manner in a different device before changing it.

I advise caution when using a standard battery charger to charge a jumpstarter via the main leads. Firstly if there is polarity protection it probably won't work and secondly the heart of the jumpstarter is a sealed lead acid battery which does not usually tolerate high charge currents without shortening the life of the battery.
This type of battery needs either the low charge current supplied by the original charger or similar or a sophisticated regulated charger.

If using a standard charger avoid using a high performance type and use only the type that provides a trickle charge.

No, you must charge the battery outside the camera using a charger. One source for a third-party charger is here.

The A3100 comes with a separate battery charger. The camera does not charge with a USB cord.

No. The PowerShots do not include the charging circuitry. That's why they come with a separate battery charger. You can buy a new charger from Canon, or a third-party charger at any reputable camera store. I can't tell you the charger model number since you didn't specify the camera model.

No, the camera does not charge the battery through the USB port. You can order a replacement battery charger from here.

Battery charger of Remington Precision MB-40 is Remington RP00043 adapter power cord replacement parts.

You can find a replacement cord on eBay very cheap. Another thing to try is to buy an external charger. Just remove the battery and place it on the charger, very helpful also is to buy another battery from the same seller. You will never run out of power with a spare battery charged.

No it won't charge through the USB port. Just buy a new generic charger for the NB-4L battery. They're very inexpensive. Here's some from Amazon:

http://www.amazon.com/s/qid=1217627331/ref=sr_st?keywords=nb-4l+charger&rs=172282&page=1&rh=n%3A172282%2Ck%3Anb-4l+charger&sort=price

See the links below for the part you are looking for.

A/C Adapter (plugs into camera)

Battery Charger (charges battery without camera)