Rechargeable batteries

The USB cable is not for charging, only for transferring. If you want to use rechargeable batteries then you must recharge them using a separate battery charger.

The VPC-S770R uses standard AA batteries. Chargers for rechargeable AA batteries should be available at just about any department, electronics, or office-supply store. Do NOT try to recharge non-disposable alkaline or lithium batteries. Also bear in mind that the camera itself is not a battery charger. If you want to use rechargeable batteries then they must be charged outside the camera using a separate battery charger.

Yes, rechargeable batteries can and will go bad. You can order a replacement battery from
http://store.kodak.com/store/ekconsus/en_US/pd/Li-Ion_Digital_Camera_Battery_KLIC-7001/baseProductID.145062700/productID.145062800

Two possibilities here. The lens housing is damage, and when you try to zoom it is unable to do so. You would be more inclined to get a message about this as opposed to the camera powering down.
Second option is a low power situation. If you are using non rechargeable batteries, you might find they are not up to the task. I have found this with few cameras. If you are using rechargeable batteries, they may be end of life, and no longer packing the same punch that the once were.
Although rechargeable batteries have 1.2 volts instead of 1.5, they can have more power with higher capacity. I tend to use 18mAh or higher batteries in my camera, especially for zooming, but make sure to use the guidelines for charging and recharging these so they will last a long time.

hello there,
You can find the Olympus C-series 5500 Sport Zoom here:
http://www.olympusamerica.com/cpg_section/cpg_support_manuals.asp?id=1151

Lithium Ion Rechargeable Battery (LI-42B)
$44.99
View Details
In Stock: Yes
from: https://emporium.olympus.com/innards/emp.asp?val100=100&step1=1318&step2=0

You can try switching over to 2500 mAh Nickel Metal Hydride rechargeable batteries and you will see a difference.
Digital cameras with zoom lens need more power than what can be supplied by ordinary batteries.

The Ni-MH batteries are recommended for the active photographer. The NI-MH batteries are rechargeable, economical and work well in low temperatures. The LB-01 CR-V3 Lithium Batteries are recommended for the less active photographer. They are lightweight, non-rechargeable batteries that provide hours of shooting.

Yes, the battery that comes with the C-7000 can be recharged. This battery is the Olympus LI-12B Lithium Ion battery and it can be recharged using the LI-10C battery charger. Both the battery and charger are included with the camera.