I was researching the symtpoms you are describing and first wanted to recommend this part of the article named iPhone: Hardware troubleshooting found here http://support.apple.com/kb/TS2802 to assist with the battery life.
- Click ( www.apple.com/batteries/iphone.html) for tips on how to prolong iPhone's battery life.
- Try turning iPhone off and then on again.
- Connect iPhone to iTunes and restore iPhone.
- If the screen shows a low-battery image, the battery is low on power and needs to charge for up to ten minutes before you can use it.
- When charging iPhone, make sure to leave it charging until it is fully charged. You'll know the battery is fully charged when the battery icon in the upper-right corner of the screen looks like this .
Note: When charging using a computer, don't connect iPhone to a keyboard. Also, the computer must be turned on and not in sleep or standby mode. If iPhone is connected to a computer that's not turned on or is in sleep or standby mode, the iPhone battery may drain.
As for the cellular connection clarity and connection, I would take a look at this iPhone Troubleshooting Assistant found here: http://www.apple.com/support/iphone/assistant/calls/