That depends on if they use the same type of ram or not. If it does you're good to go if the motherboard can handle that boost in ram. Some motherboards are built to handle only so much ram. As you stated that both are old I would check how much ram the computer was made to handle. Find the model number of the motherboard and google its stats. You might be able to accomplish this with the computer model number as well.
You may be able to get away with using the ethernet card...maybe. Does your girls computer have and ethernet slot first of all? That should be your first thing to find out. Look at the slot that it is in on the HP and see if the gateway has one just like it.
As for the hard drive as an external source you could. You can look on ebay for an external hard drive case like this one :
http://cgi.ebay.com/NEW-3-5-USB-2-0-EXTERNAL-HARD-DRIVE-ENCLOSURE-CASE-BOX_W0QQitemZ130103391610QQihZ003QQcategoryZ41911QQrdZ1QQcmdZViewItem
However for 4.3GB do you really think its worth it? My suggestion is save the cash or use your tax return and invest in a better computer.
I also know what its like to hard up for cash sometimes. So things you can do to boost the POS. First check to see if the ram will fit. 2nd buy the external case and utilize the 4.3GB hard drive as an external to free up some space on the main hard drive. Verify that the case will work with the make of the hard drive first before you buy. Ask the seller. 3rd I would get someone who knows what they doing to overclock your computer. Other than that uh if you running lime and its downloading alot of stuff end other applications or run lime overnight. Schedule your scans and updates to run overnight as well so it doesn't tie up your resources as well. Also remove unneeded programs from your startup. The only things that should run at startup is essential programs, Firewall and computer condoms ( AV Spyware stuff). 4th, run your scans on an average basis, once a week once every 3 days, after and update.
Also as for browsers, it is dependant on preference. Fire has more security built in that IE does. IE is more succeptible to spyware, adware, popups and so on. I'd go with Fire, I have no problems with anything using fire. But thats just my opinion
Just read your additional details. Yes you should be able to find drivers and stuff online. You would need model numbers of the items to find them. I would just get a new ethernet card already. You don't even know if the old one works. That should be plug and play so you won't need a driver for that. Another thing, see what the max ram your computer can handle and search online for ram. Ram has gotten cheaper now. Make sure you get the right type of Ram.