Well, in my opinion...
ArchLinux is a good lightweight distro with vanilla packages that only installs what you want. It has some incredible documentation to go with it as well. That is my lightweight distro of choice, as long as your not afraid of config files instead of gui options.
Try Wattos, it's surprisingly good.
Lubuntu, Crunchbang, or if you're more of a hardcore user, you could try an Arch or Gentoo build with Fluxbox, Openbox, or LXDE, or FreeBSD with the same WM or DE.
I'm sold to Lubuntu for lightweight. It's what I use on my low power netbook since the regular full ubuntu was too slow.
Well uhh, It depends!
For example: I have OpenSuSe 12.2 64bit with a kde desktop environment, it's memory footprint averages 950mb ram with 151 processes, However when i switch to iceWM in sessions, the average is about 230mb with 101 processes, that is a dramatic difference.
Ultimately, it's mostly the desktop environment that'll bring slower systems to a crawl...
Consider running processes as well, Google Chrome browser will require an additional 128mb of ram, that's bare minimum!
Most lite weight environments will have it's own web browser/ it's own applications to allow small footprint.
This isn't including Samba and various other Apps.
Find a distribution with what you like and choose a lite weight desktop environment, then you should be fine.
Both mint and Ubuntu have netbook remixes, try this in google:
Ubuntu netbook edition or Linux mint netbook remix.
Or try Puppy Linux