That is a bit of a slow clock speed and while the ram will serve you well enough be aware that you may have to close programs or face using the page/swap file. Similarly the low cache and that CPU might see you get the odd bit of slowdown.
Equally if that is a laptop as old as the CPU is chances are you have a 5400RPM hard drive in there- pretty much all video editing will see lots of hard drive use so consider investing in a faster one.
GPU matters little in video world right at the moment, there are a few GPU enhanced programs/decoders and filters and indeed they work well enough but they are far from essential at this point in time. On the other side of things I advise you steer clear of the GPU encoders as I have yet to see a good one.
The editors you mention are non linear editors so I assume that is a requirement (linear editors are still around but pretty much everything has at least elements of non linear editing). I am not sure why it has to be paid- there are some great freeware and open source programs out there that can go toe to toe with the big boys.
Easy to use is a different debate entirely- nothing in the video world that is easy to use will ever make anything good.
Equally I would argue there is no one be all and end all program and many people will have multiple programs depending upon what they want to do- restoration, chroma keying, transfer, basic cutting, advanced cutting (fancy transitions and what have you), audio (often considered an entirely different area to video), CGI/effects, mastering (both the video and making it into DVD, web or other formats) and whatever else you can think of.
As you asked though
freeware
avisynth- mainly windows but WINE can be used with something like ffmpegsource2 if you want (I have very little experience in such an arena though). Script driven (do not knock it until you have tried it) and incredibly powerful.
http://avisynth.org/mediawiki/Main_Page
avidemux- pretty much every platform out there
http://fixounet.free.fr/avidemux/
virtual dub- mainly windows, GUI driven
http://www.virtualdub.org/
The former and the latter have several professional grade filters and active development communities while avidemux has several filters sourced from the other two mentioned.
Blender- mainly a 3d editing/creation program but does have some serious high grade video capabilities when you need them and you can turn those 3d features to make spectacular things.
http://www.blender.org/
Back on topic as it were you do have several options
Sony's Vegas line of software:
http://www.sonycreativesoftware.com/products/vegasfamily.asp
I have not used it in over a year now but back then it was very powerful and deserving of the good name it has in the video editing circles.
Apple do have the final cut line of programs but I am not going to head down the hackintosh route with your hardware.
Cyberlink do some stuff as well and as of the last few versions became pretty nice to use. Not sure I could base my livelihood on it just yet (vegas and avisynth I would be quite happy to) but more than enough for the enthusiastic amateur.
http://www.cyberlink.com/products/powerdir...view_en_US.html
NVidia Control Panel
Some stuff else
