@MAWriterGirl The second computer is much better than the first. Its processor is four years newer, which is helpful both for the CPU itself and for the motherboard. But much more importantly, it has a dedicated graphics card. While a 1030 will likely struggle a bit with the most demanding packs (Pets, then Seasons), you should be able to get smooth play by tweaking the settings a bit. (If you want help with this process, just ask, but only when you have the computer in front of you.) The 1030 at least ten times faster in gaming than the integrated chip in the 13-year-old i5-650; to say it's not even a competition would be an understatement.
It's always a risk buying used, whether from Amazon or elsewhere, but this second one at least has a 90-day guarantee. So you'd have time to run benchmarks and stress tests and send the computer back if it failed. I can help with those too; the software you'd want to use is free and simple enough to set up.
Both these computers have more than enough storage. For the monitor, yes, you can buy a used one, it's not a problem. For the first one, you'd want a monitor that uses a VGA connection; for the second, the 1030 should have an HDMI and a DVI port. (You'd want to plug the monitor into the GPU, not the motherboard.)
For your other questions:
1. I've heard a couple people say they can't download the Super Patch, but I've had no trouble on a few different computers. It may be that the link uses http instead of https, but that doesn't mean the link itself won't work. If one browser doesn't cooperate, try another. Here's the direct download link, which points to EA servers:
http://akamai.cdn.ea.com/eadownloads/u/f/sims/sims3/patches/TS3_1.67.2.0240xx_update.exe
2. I would look into this again on the new computer, and try using Firefox. The store doesn't seem to get along with most other browsers.
3. This sounds more and more like a browser or security settings issue, and again, I'd look into it once you have the new computer set up.
4. Yes, stuff packs add only a small amount of overhead to the game, and you should be able to play with all of them at the same time. Whether you can play with most or all expansions depends on the computer you get. Like I said, you might need to make some compromises with the 1030, especially if it's the slower of the two designs Nvidia released for the card. But at worst, you might need to disable certain pets and wild horses and perhaps avoid snow; you wouldn't need to disable half your expansions.
There's nothing in your dxdiag that explains the problems you've listed other than the fact that the computer is old and has weak hardware. The dxdiag doesn't even list the processor, which could be a random glitch or a sign of a serious system issue, but I can see enough to be completely unsurprised that Sims 3 would run poorly on it. As for the downloading issues, I still suspect browser settings or perhaps an antivirus, but that's a question best left for when you're setting up a new computer anyway.
@thesims1depot I'm sorry, but I have to completely disagree with your claim that Sims 3 can run on any quad-core with 4 GB RAM. In fact, I think it's highly irresponsible to post that given how verifiably untrue it is. Windows 10 will use 2 GB RAM by itself even without any bloatware, and I can show you Sims 3 saves that use more than 2 GB just on loading. Your saves may use less, but you can't know someone else's playstyle or how long they'd like their saves to last. And Sims 3 responds very badly to having its data paged out, so it's not merely a question of letting the OS manage memory effectively.
Additionally, not all quad-core CPUs are created equal, and there are plenty of dual-cores that are more powerful than older quad-cores. As for the graphics card or chip, I have enough hwifno logs to see what happens when the GPU can't keep up with the demands of Sims 3. It's actually worse on newer systems than on old ones, which may be why you haven't seen it, but I have. One example I remember is someone who's game crashed every single time it rained or snowed—the iGPU was running at 98% capacity at the time and just couldn't meet the demands put on it, and then the game would crash. This happened even on low settings. This computer was made perhaps three or four years ago, and its iGPU was significantly better than a GT 710, although slower than a 1030.
No one is depicting Sims 3 as a AAA game needing an overpowered gaming PC. Computers with a GTX 1050 can be very cheap on Ebay or bought from friends, and a 1050 itself can be had for about $30 on Ebay.
Finally, the worst advice you can give someone on a tight budget is to buy a low-end computer that doesn't actually meet their needs. Then they've spent all of their money and don't have anything to show for it, and they may not be able to return the product either.
