MacBookPro 15" Battery/Heating Issue

I purchased my MBP in 2007 and since last summer I've been having some issues that might be related. First of all, my battery only lasts 25min or less when fully charged. I've been using it solely with power chord plugged in. Another issue that might be related to the battery issue is that my MBP gets really hot and the fanning noise gets louder especially when I'm watching youtube or hulu. After closing video websites, the noise lasts for another 10 to 15 seconds and it calms down. As for the heating issue, I feel it's the hottest near the base right below where the number keys are (so that would be between where the battery is and the base of the screen)

I've been meaning to get a new battery but I was wondering if this will solve the heating and noise issues too.

Thanks.