overall your choices are OK, I can see that you looked for the budget side of things, but that's OK. there are a few things to consider however...
1. gigabyte makes good mobos, but you should go for one with PCI-e instead of AGP. this will obviously mean getting a PCI-e video card instead, but they are often cheaper anyways. the reason being is that you may want to upgrade your video card in another year or two. currently there is still a demand for AGP cards, so they keeop making them, but this is for older systems. since you're building new, you should use the current video interface standard.
2. although the venice 3000+ is a great CPU, I bought mine when socket-939 was more expensive than it is now. currently you can get better processors for only $20-$30 more. also as sharkfood mentioned, you may want to go AM2 instead. CPU and mobo prices are similar, but DDR2 RAM is sometimes cheaper.
3. do you really need a floppy drive? i didn't know those old things even existed anymore.. hahaha
4. i don't know why you refer to the power supply as "cooling"... but anyways. aspire PSUs are junk. buy a good brand like Antec, Enermax, Fortron, PCP&C to name a few. you can find models within your price range. just to let you know, you don't need 500w either. and FYI a cheap PSU like that aspire will never be able to provide 500w.
5. if you're not going to overclock, then don't bother with the AS5 paste, the CPU will come with a cheap thermal pad/paste already on the heatsink that will work fine. if you plan to overlclock however, then you should indeed get it.
the other components don't really matter performance wise too much.
BTW, good job choosing a static wrist band, the $6 spent on that can save you hundreds of $$$ down the road
and to answer you network question, first you need to know how fast your wired internet connection is, then decide. wirelessB is 11Mb/s, wirelessG is 54Mb/s.
as an example... I use a Linksys wirelessG router and a Linksys WirelessG PCI card. my high speed broadband from comcast is 6Mb/s, so even wirelessB exceeds my fast internet connection. but wirelessG is cheap and widely available so just get that. keep in mind that the further away from the router (or the more obstructions there are), the slower the transfer speed.