https://www.kernel.org/you should use the most recent version (3.8.10).
Well I know what you meant and probably does everyone else. But I just want you to know that Linux itself is just the kernel which on it's own is pretty useless for an enduser. And therefore asking for a version of Linux is misleading.
You are searching for a GNU/Linux Distro that suits your needs. Although I do think gentoo is the best to use, it's probably not the best for you.
You want something that works out of the box so Ubuntu should be fine for you. You could also use xubuntu if you prefer it's GUI but it's basically the same.
If you don't want cannonical to sell you to amazon follow this tutorial right after installation:
http://askubuntu.com/questions/192269/how-can-i-remove-amazon-search-results-from-the-dashNow about your installation:
If your PC has a 64bit CPU use the 64bit version of course. Don't even try the 32bit version.
Are you running it natively or in a VM?
Your hardware shouldn't be a problem the only problem with hardware support you could get is 3D performance. WLAN problems are part of the past and your box should definitely boot up.
Could you please write down the reason for the kernel panic? The last time I installed ubuntu on someone's pc it messed up the partitions somehow and panicked that it couldn't find the root partition. After configuring grub myself it worked.
Also I don't think there's any use for the LTS version unless you are lazy because you can always do a release upgrade I suppose. Correct me if I'm wrong.
//edit
you are definitely using a 64bit image on a 32bit CPU. You probably messed something up during download.
Use this one for xubuntu:
http://cdimages.ubuntu.com/xubuntu/releases/raring/release/xubuntu-13.04-desktop-i386.iso Xubunto worked, the Ubuntu 64 bit said AMD in the file name and I have an intel centrino chipset so I'm assuming that is probably why I was having the problem.