Ah I remember the Compaq mystery now ,
HP & Compaq are partners or HP bought Compaq I think, so that could be it.
If you had a wireless card there would be an antenna - usually about 3" LONG OUT THE BACK OF THAT PARTICULAR CARD (darn hit caps lock LOL)

like these-
http://www.google.com/search?tbm=isc...0l3.4.3.1l11l0
Last time I talked to Verizon ( now mine is frontier) but they wanted to sell me their own wireless modem/router - so if you do go with 2 machines and don't want them both close by the dsl modem (wired) - you cloud look into the Verizon wireless modem/router.
not sure what the problems are with your internet - the not shutting down stuff - the not opening browsers....
mine was acting a lil weird so i did a clean & defrag amazingly that fixed the problem - something must have got scrambled when we had a power off /then on again right away.
you might try some scans, clean up and defrag it to see it it helps anything.