Will this CPU and graphics card handle TS3?

This computer should be able to run Sims 3, provided it's still in good shape overall, but high graphics settings may be too much to ask. There's no harm in trying though—if you're not happy with the performance, you can dial some settings back to medium or disable Pets and see how it goes.

You could replace the processor, but only with a second- or third-gen Intel CPU, so an i7-3770K at best. The socket on the motherboard doesn't support anything newer. Whether this is worth doing would depend in part on the price of a better CPU, but it may not be necessary or even helpful, depending on the graphics card you end up using: the GPU may well be the limiting factor, not the processor.

The graphics card is certainly replaceable, and in normal times you could find something else used but newer and faster for a very reasonable price. Right now, the secondhand market is still pretty outrageous, although it's getting better, so your best bet might be to wait a few months and see how this computer performs.

If and when you do want a new card, you'll need to make sure your power supply supports it. I don't know what PSU is in this system, but a GTX 650 only requires a 400W model, so that might be all you have. In that case, you wouldn't want to buy anything that required more power, but for example a 960 or 1050 ti should be fine. Be sure to check the specs before buying though, and go through a retailer that allows for returns for defective products, or if you buy from someone directly, test out the card before paying.

Start out not running anything aside from Sims 3 and see how the game performs, and you can go from there.

The only real way to tell, especially with older PSUs, is to look at the unit. It should have its essential specs printed on the side, including wattage. Some models might not show an overall figure but instead list volts and amps on each rail, but you can figure it out from that: watts = volts x amps, and the important figure is the wattage on the 12V rail, which supplies the graphics card.

If this is an older power supply, you'd also want to give it more of a buffer in terms of what your computer demands of it, and it's unfortunately not an exact science. One way to make things easier is to play with the GTX 650 in place for a while and make sure the system is stable overall, then get a new card (if and when you want one) that doesn't draw any more power than the 650. That's not too difficult—newer mid-tier GPUs have become a lot faster without really using more power; even a 1050 ti is rated to draw only 5W more than a 650, which is a trivial amount for a PSU to provide.

I'm glad the system is performing well now, but adding even mid-tier expansion packs will significantly increase Sims 3's demands. So it's still a good idea to add one or two at a time and see how things go.

It will also be helpful to manually cap in-game framerates, which you can do through the built-in Nvidia Control Panel. This limits the graphics card's workload and can also smooth out some graphical issues that can show up at higher fps. Right-click on the desktop, select the CP, then Manage 3D Settings > Program Settings, then ts3.exe (Origin install) or ts3w.exe (with the W, disc or Steam install). If you play in fullscreen mode all the time, set Vertical sync to Adaptive, which will cap fps to your monitor's refresh rate, probably 60 Hz, without further lowering fps when it's below that level. And enable Triple Buffering. If you ever play in windowed mode, set Max Frame Rate to your monitor's refresh rate as well; this setting will apply in both fullscreen and windowed modes but won't give the other benefits of v-sync.

You can see in-game framerates by bringing up the cheats console and entering "fps on" without quotes. The number will appear in the upper right corner of the screen. To find your monitor's refresh rate, right-click on the desktop and select Display Settings > Advanced Display Settings; the value will be listed in Hz and is probably 60, but it could be higher if your monitor is relatively new. If it happens to be 120 or 144, it would be better for your graphics card to set vertical sync to "Adaptive: half refresh rate" to cap it to 60 of 72, and/or to set the max frame rate to the same.

@Goosey I did not tell the OP to purchase any new processor, I said they only could upgrade to a high-end 3rd-gen Intel CPU at best, at least without replacing the motherboard. And I also said it might not be necessary depending on how their system was handling Sims 3 and might not be worth the price anyway.

Additionally, I don't know if you noticed, but they have a non-K CPU, unlike you. The i7-2600 non-K can only be overclocked to a very limited degree, doing so is much more complicated than with the K variant, and their motherboard might not even support it. A 3770K could be a significant improvement, especially if overclocking was necessary for good performance, although again that's not at all a foregone conclusion.

As for heat, there are plenty of CPU coolers that have no trouble handling a 3rd-gen CPU with or without an overclock. The amount of heat generated is only an issue if the cooler or the case airflow can't keep up; it's never by itself a reason not to buy a particular processor. The 3rd-gen models do NOT thermal throttle at default clock speeds if the cooler is doing its job, and casually throwing around misleading information like that is the kind of thing that helps people make bad purchasing decisions.

The GTX 950 can run all Sims 3 packs together, likely on high settings if not ultra. It's a mid-range card and was always meant to be one, there's no shame in that and it's not "nerfed" relative to its original purpose or price point. The higher model cards are of course faster, that's the whole point, but they also require more power and you can't just stick one in a system that isn't built to handle it. But at the moment, the point is moot, since the OP has a 650 and hasn't decided whether an upgrade is necessary or warranted.

In my experience, yes that graphics card should run it just fine. However, when I first got my computer (also an Nvidia card), I had to actually manually set a framerate cap for TS3 because it kept trying to run at 1000+ FPS. That was the only issue, which in turn caused things like lag and even my laptop to overheat. Setting a framerate cap of 60 fixed it, and now things run without my computer even having to try.

But, if you don't have any issues, then you don't need to worry about it! That's great it runs! :)

The GTX 650 was already slow in 2012 ...

Thank you for your insight @puzzlezaddict! I just tried the base game and it runs incredibly fast and smooth. The computer wasn't struggling at all and there wasn't any lag even with reflections and high render distance / Sim and texture detail (I did leave other settings on medium and turned off high detailed lots). I'm optimistic it will be able to handle the rest of the packs pretty well too, although I will probably leave Pets disabled just to be on the safe side.

If need be, I'll keep an eye on those graphics card you recommended. Quick question: how do you find out what kind of PSU you have?