Hi, don't worry lol, compared to some of my mates your a rocket scientist lol, one mate starts to wail at the point I say press the power button lol
Right with the info so far I would say you are spot on with the ram, BUT, there are 2 points that can\could change this, and that comes back to the limitations of the hardware itself (it MUST be 64bit capable), and the operating system (it MUST be a 64bit OS to use 4gig + of ram), you said you have windows 7 on there now, but not which version it is, but to be honest given the crucial scan, and probable age of the system, I tend to doubt it could handle 64bit, but I would like to identify the motherboard before saying much more on the ram.
So treating this as 32bit, I agree that you would probably be best served getting 2x1gig of ram, but I would like to confirm the board can take 1gig sticks, as this can also be a hardware limitation of the motherboard directly, there are, or at least were, boards that could only take a max of 512meg in each slot, and as the scan didn't even mention 1gig sticks (the term for ram, as opposed to cards

), and that concerns me, I would like to try and ensure the board could take 1gig sticks before saying they will work.
So if you can go
Here and download\install Speccy, then run it, we can find out a bit more with luck, on the left once you have launched it you will have various headings, click on the one that says Motherboard, then on the right it will give a load of info about it, if you can report back with the make and model of the board, I can then try and find out more about it, but the problem is often the makers of systems (Sony in this case) often pay hardware makers to produce a limited run on rebadged boards, where if you search the board alone, you can find little about it, as the board name is exclusive to that unit, but fingers crossed it has enough for us to get what we need.
Now the hard drive, given you have Windows 7 on there, it's probable that it should see the whole of a large drive, creating a partition is probably worst case here, but I mention it to raise awareness, so install the hard drive, and fire the system up (I will guide how to shortly), it should pick it up as a new device, and a wizard should open and say it needs formatting before it can be used, just accept all it throws at you and next it through, and then the drive should be visible\useable under your My Computer, so once installed go to there and see what the drive is reported as, if it shows as the full size, your laughing, you don't need do anything more, if it only shows as a lesser size, then it would need to have a partition(s) created to use the full drive.
The hard drive(s) you mention can be bettered, the ST2000DM001 has a faster spin speed, so it "should" be faster, as the best I can make out it is the updated version of the one you mention, but others who know hardware better than I may know better, and be able to suggest something even better, but 2TB is the max I would suggest for that system, once you hit 3TB the motherboards bios version comes into play, and with the age of it I wouldn't be comfortable in saying such a big drive would work on it, but I take it the idea is to have this as an internal drive (which is what I have based my info on so far), as opposed to another external one ?
But for peace of mind open the side (using guide below concerning power), and check on the motherboard visually that there are spare Sata ports, if there are none spare then the drive will need an additional card before you can connect it, the ports will be in a cluster of 2, 4 or possibly more depending on the board, the existing hard drive will be connected to one of these already, so look for a thin lead, commonly red, but that does vary, but if you see spare ports then you are good to go, but remember that number, as that is the limit of Sata devices that board can manage.
When installing ANYTHING internally you must be very careful, first shut the system down (if powered on), turn the PSU switch to off, and remove the mains lead from the tower (belt and braces approach), only after doing this you are safe to remove the side panel.
Static electricity is the enemy when working with computer parts, and the ram is among the highest at risk, and a hard drive is lesser, you can buy anti static devices, anything from a crocodile clip that clips to the case with Velcro securing it to you, to a full on mat that plugs into the mains, not to use live power, purely to use the earthing of the ring main (assuming your country uses an earth in house wiring ?), but at a minimum once you are sat comfortably ready to do the work, touch the actual case prior to touching ANYTHING inside, or the parts your adding (more so with the ram when you get it), and never touch the circuit board or any gold\brass contacts (thinking of the ram here more), and remember it only takes a couple of steps (talking about walking here) to generate more static, so if you need to move, touch the case before continuing again.
Next lay the system on it's side for ease of access, and identify the bay you want to use (there is no difference which space in the cage you use), and secure the drive into there using the screws supplied with it, anything longer can damage the drive, , then add the power and Sata leads to it, and connect the other end of the Sata lead to the free Sata port on the motherboard (there may be numerous, in which case any will be fine), and ensure the leads are laid as neatly as you can for max air flow for cooling, and check they are securely fixed, and take note about the Sata leads, if they have a kind of metal "clip" on one side of the plug, you must depress that bit of metal (press it into the plug so it becomes flush if you follow) before pulling it out of the socket\port, it's like a locking clip, just incase you need to ever remove one.
Once your happy all is well, replace the side, stand it up, plug in the power, and switch it on at the back of the PSU, now power the system on and use as normal, if all is well you should get the wizard come up as I mentioned earlier in this manual lol, so although I have written a lot here, just take it a paragraph at a time, I tried to address each point in it's own space to avoid confusion, but there is some crossover as I addressed various stages, if your stuck on any of this please just say, and I will try and clarify the bit your stuck on if needed, hope this helps you a bit anyway.