Which processor should I use in my new computer?

Any of those processors would be fine for Sims 3. If you want to futureproof your computer a bit though, get the Ryzen 5 3600. It's much stronger than the 3200G and 3400G (and outperforms the 2700 as well), so it will be able to keep up with new apps, including games, for longer than the others. It also means that if you want, you'll be able to swap out the graphics card in a few years without worrying about whether the CPU can keep up. There's not much point in getting the 3600X though, unless you plan to overclock your system at some point.

The other specs are all good or great for TS3. The 1660 is overkill—you'd be more than fine with a 1650. But you might want the faster card for other purposes, like other games.

If you have a bit more money to spare, upgrading the SSD to a 512 GB model would be a good investment. A 256 GB drive is enough for Sims 3 plus all of your user content, but it still might be somewhat of a tight fit. So you might end up needing to install other apps to the HDD, or shift other files back and forth to keep enough space free. You wouldn't need to worry about that with a 512 GB drive, at least up to a point.

It might be nice to have more memory too, but it's certainly not necessary for TS3. RAM is cheap right now though, so it's worth considering. Getting 16 GB would make multitasking easier, especially if you like having browser windows open while you play.

I'm also curious about the power supply. You didn't mention it, but in many ways, it's the single most important component. Getting a cheap model can negatively impact your entire system, even if everything else is working smoothly.

Still, this is a very good computer for Sims 3, even as is. You shouldn't have any trouble playing on ultra graphics settings, beyond the limits of the game itself.

Air cooling will be fine for any of the processors, although I'd want to get a high-end version (not the stock AMD cooler) for the 3600X if you do happen to get that CPU. Otherwise, the stock cooler will be good enough.

For the power supply, it's better to get at least a bronze-rated model from a reputable company. (Cooler Master is fine.) 400W should be enough, even if Nvidia recommends a 450W PSU for a 1660. The calculators I tried all said 400W would cover all the hardware you listed. Just be aware that if you decide to upgrade at some point, you might need to get a new PSU as well, depending on the component being upgraded.

There aren't any bronze or gold rated PSUs in that category? Thermaltake is fine too, but the extra wattage here isn't really going to make a difference right away. However, it might help a bit a few years down the road, when the power supply is a bit older and starting to waver.

Some high-end PSUs come with ten year guarantees, but even those models can get a bit unreliable towards the end. The ones you've listed likely won't last that long, but they should be perfectly fine for several years at the very least. Many can and do last longer, but they're not always as reliable at that point, especially if your definition of reliable involves playing games for long stretches. The reliability is a factor of the rating though, not so much the wattage. (As always with hardware, there's some element of randomness to how well a component holds up over time.) Where the higher wattage can help is when the PSU is just beginning to falter, and you'd like to wait a bit longer to replace it.

So it's kind of a close call. Personally, I'd spend a little extra for the upgrade, especially if the amount was trivial. Having a larger margin for error is always nice. But it's not strictly necessary—the Cooler Master should still be good enough for a while, and by the time it's not, you'd probably be about ready a new PSU whether you bought that or the Thermaltake.

Thank you @puzzlezaddict! The power supply is a Cooler Master MWE 400W 80+ Power Supply. I’m going to add that I am going to use air cooling in my system.

The configurator I’m using (I’m not building it myself) offers a Thermaltake 500W 80+ Power Supply. Should I go for that one instead?

They Don’t offer any Bronze rated PSUs, and the Gold rated ones are 600 Watts or above.

Then I think you'll be fine with the Cooler Master, or the Thermaltake if you prefer. Just keep in mind that you'll probably need to replace it if you do any upgrades later.

Ok, thank you.